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	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pos_real_is_real_pos
2	instantiation	4, 5, 6	,  ⊢
	: , : , :
3	instantiation	7, 232, 131, 18, 132, 28, 8, 9, 10*	,  ⊢
	: , : , : , : , : , :
4	theorem		⊢
	proveit.logic.equality.sub_right_side_into
5	instantiation	150, 11, 20	,  ⊢
	: , :
6	instantiation	53, 131, 232, 225, 132, 18, 215, 19, 12	,  ⊢
	: , : , : , : , : , :
7	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_factor_bound
8	instantiation	230, 39, 13	⊢
	: , : , :
9	instantiation	14, 15, 16	,  ⊢
	: , : , :
10	instantiation	17, 232, 131, 18, 132, 215, 19	⊢
	: , : , : , :
11	instantiation	150, 221, 25	⊢
	: , :
12	instantiation	230, 220, 20	,  ⊢
	: , : , :
13	instantiation	230, 201, 36	⊢
	: , : , :
14	theorem		⊢
	proveit.numbers.ordering.transitivity_less_less_eq
15	instantiation	114, 21	,  ⊢
	:
16	instantiation	22, 23, 72, 24*	,  ⊢
	:
17	theorem		⊢
	proveit.numbers.multiplication.mult_zero_any
18	instantiation	212	⊢
	: , :
19	instantiation	230, 220, 25	⊢
	: , : , :
20	instantiation	58, 101, 198, 26	,  ⊢
	: , : , :
21	instantiation	27, 28, 29	,  ⊢
	: , :
22	theorem		⊢
	proveit.trigonometry.sine_linear_bound_nonneg
23	instantiation	30, 49, 31	,  ⊢
	:
24	instantiation	187, 32, 33	⊢
	: , : , :
25	instantiation	34, 35, 36	⊢
	: , : , :
26	instantiation	37, 38	,  ⊢
	:
27	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
28	instantiation	230, 39, 186	⊢
	: , : , :
29	instantiation	40, 41	,  ⊢
	:
30	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonneg_real_is_real_nonneg
31	instantiation	42, 60	,  ⊢
	: , :
32	instantiation	165, 43	⊢
	: , : , :
33	instantiation	187, 44, 45	⊢
	: , : , :
34	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
35	instantiation	46, 47	⊢
	: , :
36	theorem		⊢
	proveit.physics.quantum.QPE._two_pow_t_is_nat_pos
37	theorem		⊢
	proveit.trigonometry.sine_pos_interval
38	instantiation	48, 101, 182, 49, 50	,  ⊢
	: , : , :
39	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
40	theorem		⊢
	proveit.numbers.absolute_value.abs_nonzero_closure
41	instantiation	51, 139, 52	,  ⊢
	:
42	theorem		⊢
	proveit.numbers.ordering.relax_less
43	instantiation	53, 225, 232, 131, 54, 132, 215, 168, 95	⊢
	: , : , : , : , : , :
44	instantiation	187, 55, 56	⊢
	: , : , :
45	instantiation	57, 68	⊢
	:
46	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
47	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
48	theorem		⊢
	proveit.numbers.number_sets.real_numbers.in_IntervalOO
49	instantiation	58, 101, 85, 83	,  ⊢
	: , : , :
50	instantiation	59, 60, 61	,  ⊢
	: , :
51	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.nonzero_complex_is_complex_nonzero
52	instantiation	62, 63	,  ⊢
	: , :
53	theorem		⊢
	proveit.numbers.multiplication.disassociation
54	instantiation	212	⊢
	: , :
55	instantiation	165, 64	⊢
	: , : , :
56	instantiation	65, 66, 67, 68, 69*	⊢
	: , : , :
57	theorem		⊢
	proveit.numbers.division.frac_one_denom
58	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
59	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
60	instantiation	70, 101, 85, 83	,  ⊢
	: , : , :
61	instantiation	71, 72, 73	,  ⊢
	: , : , :
62	theorem		⊢
	proveit.numbers.addition.subtraction.nonzero_difference_if_different
63	instantiation	74, 75, 76, 99	,  ⊢
	: , :
64	instantiation	187, 77, 78	⊢
	: , : , :
65	theorem		⊢
	proveit.numbers.division.frac_cancel_left
66	instantiation	230, 192, 79	⊢
	: , : , :
67	instantiation	230, 192, 80	⊢
	: , : , :
68	instantiation	230, 220, 81	⊢
	: , : , :
69	instantiation	214, 168	⊢
	:
70	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
71	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less
72	instantiation	82, 101, 85, 83	,  ⊢
	: , : , :
73	instantiation	84, 85, 86, 142, 87, 88*, 89*	⊢
	: , : , :
74	theorem		⊢
	proveit.physics.quantum.QPE._delta_b_not_eq_scaledNonzeroInt
75	instantiation	90, 131, 225, 132	⊢
	: , : , : , : , :
76	instantiation	230, 91, 164	⊢
	: , : , :
77	instantiation	92, 131, 225, 132, 215, 168, 95	⊢
	: , : , : , : , : , : , :
78	instantiation	93, 225, 232, 131, 94, 132, 168, 215, 95	⊢
	: , : , : , : , : , :
79	instantiation	230, 96, 196	⊢
	: , : , :
80	instantiation	230, 208, 97	⊢
	: , : , :
81	instantiation	150, 221, 112	⊢
	: , :
82	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_upper_bound
83	instantiation	98, 164, 99	,  ⊢
	:
84	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
85	instantiation	197, 182, 221, 199	⊢
	: , :
86	instantiation	150, 151, 101	⊢
	: , :
87	instantiation	100, 151, 101, 182, 102, 103	⊢
	: , : , :
88	instantiation	104, 105, 106, 107	⊢
	: , : , : , :
89	instantiation	187, 108, 109	⊢
	: , : , :
90	theorem		⊢
	proveit.logic.sets.enumeration.in_enumerated_set
91	instantiation	110, 111, 137	⊢
	: , :
92	theorem		⊢
	proveit.numbers.multiplication.leftward_commutation
93	theorem		⊢
	proveit.numbers.multiplication.association
94	instantiation	212	⊢
	: , :
95	instantiation	230, 220, 112	⊢
	: , : , :
96	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
97	instantiation	230, 217, 113	⊢
	: , : , :
98	theorem		⊢
	proveit.physics.quantum.QPE._scaled_abs_delta_b_floor_diff_interval
99	assumption		⊢
100	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_right_factor_bound
101	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
102	instantiation	114, 196	⊢
	:
103	instantiation	115, 171	⊢
	:
104	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
105	instantiation	187, 116, 117	⊢
	: , : , :
106	instantiation	118	⊢
	:
107	instantiation	119, 141	⊢
	: , :
108	instantiation	165, 141	⊢
	: , : , :
109	instantiation	119, 120, 121*	⊢
	: , :
110	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
111	instantiation	122, 123, 219	⊢
	: , :
112	instantiation	230, 124, 125	⊢
	: , : , :
113	instantiation	230, 223, 200	⊢
	: , : , :
114	theorem		⊢
	proveit.numbers.number_sets.real_numbers.positive_if_in_real_pos
115	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
116	instantiation	187, 126, 127	⊢
	: , : , :
117	instantiation	128, 129	⊢
	:
118	axiom		⊢
	proveit.logic.equality.equals_reflexivity
119	theorem		⊢
	proveit.logic.equality.equals_reversal
120	instantiation	130, 131, 232, 225, 132, 133, 169, 168	⊢
	: , : , : , : , : , :
121	instantiation	187, 134, 135	⊢
	: , : , :
122	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
123	instantiation	136, 137	⊢
	:
124	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_nonneg_within_real
125	instantiation	138, 139	⊢
	:
126	instantiation	165, 140	⊢
	: , : , :
127	instantiation	165, 141	⊢
	: , : , :
128	theorem		⊢
	proveit.numbers.addition.elim_zero_left
129	instantiation	230, 220, 142	⊢
	: , : , :
130	theorem		⊢
	proveit.numbers.multiplication.distribute_through_sum
131	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
132	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
133	instantiation	212	⊢
	: , :
134	instantiation	165, 143	⊢
	: , : , :
135	instantiation	213, 168	⊢
	:
136	theorem		⊢
	proveit.numbers.negation.int_closure
137	instantiation	230, 144, 145	⊢
	: , : , :
138	theorem		⊢
	proveit.numbers.absolute_value.abs_complex_closure
139	instantiation	230, 220, 146	⊢
	: , : , :
140	instantiation	147, 169	⊢
	:
141	instantiation	148, 168, 215, 199, 149*	⊢
	: , :
142	instantiation	150, 151, 182	⊢
	: , :
143	instantiation	152, 219, 229, 153*	⊢
	: , : , : , :
144	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
145	theorem		⊢
	proveit.physics.quantum.QPE._two_pow_t_minus_one_is_nat_pos
146	instantiation	154, 155, 183, 156	⊢
	: , : , :
147	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
148	theorem		⊢
	proveit.numbers.division.div_as_mult
149	instantiation	187, 157, 158	⊢
	: , : , :
150	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
151	instantiation	230, 226, 159	⊢
	: , : , :
152	theorem		⊢
	proveit.numbers.addition.rational_pair_addition
153	instantiation	187, 160, 161	⊢
	: , : , :
154	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_co__is__real
155	instantiation	162, 183	⊢
	:
156	instantiation	163, 164	⊢
	:
157	instantiation	165, 166	⊢
	: , : , :
158	instantiation	167, 168, 169	⊢
	: , :
159	instantiation	230, 170, 171	⊢
	: , : , :
160	instantiation	202, 232, 172, 173, 174, 175	⊢
	: , : , : , :
161	instantiation	176, 177, 178	⊢
	:
162	theorem		⊢
	proveit.numbers.negation.real_closure
163	theorem		⊢
	proveit.physics.quantum.QPE._delta_b_floor_diff_in_interval
164	assumption		⊢
165	axiom		⊢
	proveit.logic.equality.substitution
166	instantiation	179, 180, 200, 181*	⊢
	: , :
167	theorem		⊢
	proveit.numbers.multiplication.commutation
168	instantiation	230, 220, 182	⊢
	: , : , :
169	instantiation	230, 220, 183	⊢
	: , : , :
170	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
171	instantiation	184, 185, 186	⊢
	: , :
172	instantiation	212	⊢
	: , :
173	instantiation	212	⊢
	: , :
174	instantiation	187, 188, 189	⊢
	: , : , :
175	theorem		⊢
	proveit.numbers.numerals.decimals.mult_2_2
176	theorem		⊢
	proveit.numbers.division.frac_cancel_complete
177	instantiation	230, 220, 190	⊢
	: , : , :
178	instantiation	211, 191	⊢
	:
179	theorem		⊢
	proveit.numbers.exponentiation.neg_power_as_div
180	instantiation	230, 192, 193	⊢
	: , : , :
181	instantiation	194, 215	⊢
	:
182	instantiation	230, 195, 196	⊢
	: , : , :
183	instantiation	197, 198, 221, 199	⊢
	: , :
184	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
185	instantiation	230, 201, 200	⊢
	: , : , :
186	instantiation	230, 201, 224	⊢
	: , : , :
187	axiom		⊢
	proveit.logic.equality.equals_transitivity
188	instantiation	202, 232, 203, 204, 205, 206	⊢
	: , : , : , :
189	theorem		⊢
	proveit.numbers.numerals.decimals.add_2_2
190	instantiation	230, 226, 207	⊢
	: , : , :
191	theorem		⊢
	proveit.numbers.numerals.decimals.posnat4
192	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
193	instantiation	230, 208, 209	⊢
	: , : , :
194	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
195	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
196	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pi_is_real_pos
197	theorem		⊢
	proveit.numbers.division.div_real_closure
198	instantiation	230, 226, 210	⊢
	: , : , :
199	instantiation	211, 224	⊢
	:
200	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
201	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
202	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
203	instantiation	212	⊢
	: , :
204	instantiation	212	⊢
	: , :
205	instantiation	213, 215	⊢
	:
206	instantiation	214, 215	⊢
	:
207	instantiation	230, 228, 216	⊢
	: , : , :
208	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
209	instantiation	230, 217, 218	⊢
	: , : , :
210	instantiation	230, 228, 219	⊢
	: , : , :
211	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
212	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
213	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
214	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
215	instantiation	230, 220, 221	⊢
	: , : , :
216	instantiation	230, 231, 222	⊢
	: , : , :
217	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
218	instantiation	230, 223, 224	⊢
	: , : , :
219	instantiation	230, 231, 225	⊢
	: , : , :
220	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
221	instantiation	230, 226, 227	⊢
	: , : , :
222	theorem		⊢
	proveit.numbers.numerals.decimals.nat4
223	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
224	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
225	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
226	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
227	instantiation	230, 228, 229	⊢
	: , : , :
228	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
229	instantiation	230, 231, 232	⊢
	: , : , :
230	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
231	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
232	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
*equality replacement requirements