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	⊢
	: , :
1	reference	183	⊢
2	instantiation	5, 6	⊢
	:
3	reference	12	⊢
4	instantiation	7, 8	⊢
	: , :
5	theorem		⊢
	proveit.trigonometry.real_closure
6	instantiation	69, 9, 10	⊢
	: , : , :
7	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
8	instantiation	11, 41, 12, 13	⊢
	: , :
9	instantiation	97, 14, 110	⊢
	: , :
10	instantiation	73, 128, 222, 215, 129, 15, 133, 86, 87	⊢
	: , : , : , : , : , :
11	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq
12	instantiation	16, 41, 184, 18	⊢
	: , : , :
13	instantiation	17, 41, 184, 18	⊢
	: , : , :
14	instantiation	97, 160, 109	⊢
	: , :
15	instantiation	203	⊢
	: , :
16	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
17	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
18	instantiation	19, 20	⊢
	:
19	theorem		⊢
	proveit.trigonometry.sine_pos_interval
20	instantiation	21, 41, 109, 22, 23	⊢
	: , : , :
21	theorem		⊢
	proveit.numbers.number_sets.real_numbers.in_IntervalOO
22	instantiation	220, 136, 27	⊢
	: , : , :
23	instantiation	24, 25, 26	⊢
	: , :
24	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
25	instantiation	96, 27	⊢
	:
26	instantiation	28, 29, 30	⊢
	: , : , :
27	instantiation	31, 135, 137	⊢
	: , :
28	axiom		⊢
	proveit.numbers.ordering.transitivity_less_less
29	instantiation	69, 32, 50	⊢
	: , : , :
30	instantiation	80, 78, 33, 34, 35*, 36*	⊢
	: , : , :
31	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
32	instantiation	37, 38, 39	⊢
	: , : , :
33	instantiation	97, 98, 41	⊢
	: , :
34	instantiation	40, 98, 41, 109, 72, 42	⊢
	: , : , :
35	instantiation	170, 43, 44	⊢
	: , : , :
36	instantiation	45, 46, 47*	⊢
	: , :
37	theorem		⊢
	proveit.numbers.ordering.transitivity_less_less_eq
38	instantiation	48, 49, 50	⊢
	: , : , :
39	instantiation	51, 109, 52, 165, 53, 54, 55*, 56*	⊢
	: , : , :
40	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_right_factor_bound
41	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
42	instantiation	57, 147	⊢
	:
43	instantiation	111, 58	⊢
	: , : , :
44	instantiation	59, 60	⊢
	:
45	theorem		⊢
	proveit.logic.equality.equals_reversal
46	instantiation	61, 128, 222, 215, 129, 62, 77, 86	⊢
	: , : , : , : , : , :
47	instantiation	170, 63, 64	⊢
	: , : , :
48	theorem		⊢
	proveit.logic.equality.sub_left_side_into
49	instantiation	65, 215, 135, 137, 66, 67*	⊢
	: , : , : , : , : , :
50	instantiation	111, 68	⊢
	: , : , :
51	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
52	instantiation	97, 160, 110	⊢
	: , :
53	instantiation	69, 70, 71	⊢
	: , : , :
54	instantiation	144, 72	⊢
	: , :
55	instantiation	73, 215, 222, 128, 74, 129, 86, 133, 87	⊢
	: , : , : , : , : , :
56	instantiation	75, 86, 77	⊢
	: , :
57	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
58	instantiation	76, 77	⊢
	:
59	theorem		⊢
	proveit.numbers.addition.elim_zero_left
60	instantiation	220, 212, 78	⊢
	: , : , :
61	theorem		⊢
	proveit.numbers.multiplication.distribute_through_sum
62	instantiation	203	⊢
	: , :
63	instantiation	111, 79	⊢
	: , : , :
64	instantiation	204, 86	⊢
	:
65	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_factor_bound
66	instantiation	80, 159, 213, 81, 82, 83*, 84*	⊢
	: , : , :
67	instantiation	85, 215, 86, 87	⊢
	: , : , : , :
68	instantiation	170, 88, 89	⊢
	: , : , :
69	theorem		⊢
	proveit.logic.equality.sub_right_side_into
70	instantiation	90, 91, 92	⊢
	: , :
71	instantiation	93, 202, 94, 133, 181, 95*	⊢
	: , :
72	instantiation	96, 135	⊢
	:
73	theorem		⊢
	proveit.numbers.multiplication.disassociation
74	instantiation	203	⊢
	: , :
75	theorem		⊢
	proveit.numbers.multiplication.commutation
76	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
77	instantiation	220, 212, 165	⊢
	: , : , :
78	instantiation	97, 98, 109	⊢
	: , :
79	instantiation	99, 211, 219, 100*	⊢
	: , : , : , :
80	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
81	instantiation	220, 216, 101	⊢
	: , : , :
82	instantiation	102, 213, 160, 184, 103, 104	⊢
	: , : , :
83	instantiation	105, 139, 206, 106	⊢
	: , : , :
84	instantiation	170, 107, 108	⊢
	: , : , :
85	theorem		⊢
	proveit.numbers.multiplication.elim_one_any
86	instantiation	220, 212, 109	⊢
	: , : , :
87	instantiation	220, 212, 110	⊢
	: , : , :
88	instantiation	111, 112	⊢
	: , : , :
89	instantiation	113, 181	⊢
	:
90	theorem		⊢
	proveit.numbers.absolute_value.weak_upper_bound
91	instantiation	114, 142, 165, 143	⊢
	: , : , :
92	instantiation	144, 115	⊢
	: , :
93	theorem		⊢
	proveit.numbers.absolute_value.abs_prod
94	instantiation	203	⊢
	: , :
95	instantiation	116, 117	⊢
	:
96	theorem		⊢
	proveit.numbers.number_sets.real_numbers.positive_if_in_real_pos
97	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
98	instantiation	220, 216, 118	⊢
	: , : , :
99	theorem		⊢
	proveit.numbers.addition.rational_pair_addition
100	instantiation	170, 119, 120	⊢
	: , : , :
101	instantiation	220, 218, 121	⊢
	: , : , :
102	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right_strong
103	instantiation	122, 213, 184, 123, 124, 125, 126*	⊢
	: , : , :
104	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
105	theorem		⊢
	proveit.numbers.addition.subtraction.subtract_from_add
106	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
107	instantiation	127, 128, 222, 215, 129, 130, 133, 139, 131	⊢
	: , : , : , : , : , :
108	instantiation	132, 139, 133, 134	⊢
	: , : , :
109	instantiation	220, 136, 135	⊢
	: , : , :
110	instantiation	220, 136, 137	⊢
	: , : , :
111	axiom		⊢
	proveit.logic.equality.substitution
112	instantiation	138, 139, 181, 140*	⊢
	: , :
113	theorem		⊢
	proveit.numbers.absolute_value.abs_even
114	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_co_lower_bound
115	instantiation	141, 142, 165, 143	⊢
	: , : , :
116	theorem		⊢
	proveit.numbers.absolute_value.abs_non_neg_elim
117	instantiation	144, 145	⊢
	: , :
118	instantiation	220, 146, 147	⊢
	: , : , :
119	instantiation	188, 222, 148, 149, 150, 151	⊢
	: , : , : , :
120	instantiation	152, 153, 154	⊢
	:
121	instantiation	220, 221, 155	⊢
	: , : , :
122	theorem		⊢
	proveit.numbers.exponentiation.exp_monotonicity_large_base_less_eq
123	instantiation	178, 179, 157	⊢
	: , : , :
124	theorem		⊢
	proveit.numbers.numerals.decimals.less_1_2
125	instantiation	156, 157	⊢
	:
126	instantiation	158, 206	⊢
	:
127	theorem		⊢
	proveit.numbers.addition.disassociation
128	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
129	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
130	instantiation	203	⊢
	: , :
131	instantiation	220, 212, 159	⊢
	: , : , :
132	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_23
133	instantiation	220, 212, 160	⊢
	: , : , :
134	instantiation	161	⊢
	:
135	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pi_is_real_pos
136	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
137	instantiation	162, 163	⊢
	:
138	theorem		⊢
	proveit.numbers.multiplication.mult_neg_left
139	instantiation	220, 212, 184	⊢
	: , : , :
140	instantiation	204, 181	⊢
	:
141	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_co_upper_bound
142	instantiation	164, 165	⊢
	:
143	theorem		⊢
	proveit.physics.quantum.QPE._scaled_delta_b_round_in_interval
144	theorem		⊢
	proveit.numbers.ordering.relax_less
145	instantiation	166, 195	⊢
	:
146	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
147	instantiation	167, 168, 169	⊢
	: , :
148	instantiation	203	⊢
	: , :
149	instantiation	203	⊢
	: , :
150	instantiation	170, 171, 172	⊢
	: , : , :
151	theorem		⊢
	proveit.numbers.numerals.decimals.mult_2_2
152	theorem		⊢
	proveit.numbers.division.frac_cancel_complete
153	instantiation	220, 212, 173	⊢
	: , : , :
154	instantiation	201, 174	⊢
	:
155	instantiation	175, 176, 215	⊢
	: , :
156	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_lower_bound
157	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
158	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
159	instantiation	220, 216, 177	⊢
	: , : , :
160	instantiation	178, 179, 195	⊢
	: , : , :
161	axiom		⊢
	proveit.logic.equality.equals_reflexivity
162	theorem		⊢
	proveit.numbers.absolute_value.abs_nonzero_closure
163	instantiation	180, 181, 182	⊢
	:
164	theorem		⊢
	proveit.numbers.negation.real_closure
165	instantiation	183, 184, 213, 185	⊢
	: , :
166	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
167	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
168	instantiation	220, 187, 186	⊢
	: , : , :
169	instantiation	220, 187, 202	⊢
	: , : , :
170	axiom		⊢
	proveit.logic.equality.equals_transitivity
171	instantiation	188, 222, 189, 190, 191, 192	⊢
	: , : , : , :
172	theorem		⊢
	proveit.numbers.numerals.decimals.add_2_2
173	instantiation	220, 216, 193	⊢
	: , : , :
174	theorem		⊢
	proveit.numbers.numerals.decimals.posnat4
175	theorem		⊢
	proveit.numbers.addition.add_nat_closure_bin
176	instantiation	220, 194, 195	⊢
	: , : , :
177	instantiation	220, 218, 196	⊢
	: , : , :
178	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
179	instantiation	197, 198	⊢
	: , :
180	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.nonzero_complex_is_complex_nonzero
181	instantiation	220, 212, 199	⊢
	: , : , :
182	assumption		⊢
183	theorem		⊢
	proveit.numbers.division.div_real_closure
184	instantiation	220, 216, 200	⊢
	: , : , :
185	instantiation	201, 202	⊢
	:
186	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
187	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
188	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
189	instantiation	203	⊢
	: , :
190	instantiation	203	⊢
	: , :
191	instantiation	204, 206	⊢
	:
192	instantiation	205, 206	⊢
	:
193	instantiation	220, 218, 207	⊢
	: , : , :
194	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
195	theorem		⊢
	proveit.physics.quantum.QPE._two_pow_t_is_nat_pos
196	instantiation	208, 211	⊢
	:
197	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
198	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
199	instantiation	209, 210	⊢
	:
200	instantiation	220, 218, 211	⊢
	: , : , :
201	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
202	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
203	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
204	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
205	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
206	instantiation	220, 212, 213	⊢
	: , : , :
207	instantiation	220, 221, 214	⊢
	: , : , :
208	theorem		⊢
	proveit.numbers.negation.int_closure
209	theorem		⊢
	proveit.physics.quantum.QPE._delta_b_is_real
210	theorem		⊢
	proveit.physics.quantum.QPE._best_round_is_int
211	instantiation	220, 221, 215	⊢
	: , : , :
212	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
213	instantiation	220, 216, 217	⊢
	: , : , :
214	theorem		⊢
	proveit.numbers.numerals.decimals.nat4
215	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
216	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
217	instantiation	220, 218, 219	⊢
	: , : , :
218	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
219	instantiation	220, 221, 222	⊢
	: , : , :
220	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
221	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
222	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
*equality replacement requirements