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