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.number_sets.real_numbers.in_IntervalOO
2	reference	190	⊢
3	reference	199	⊢
4	instantiation	132, 47, 50	,  ⊢
	: , :
5	instantiation	6, 7, 8	,  ⊢
	: , :
6	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
7	instantiation	11, 9, 10	,  ⊢
	: , : , :
8	instantiation	11, 12, 13	,  ⊢
	: , : , :
9	instantiation	49, 95, 14, 50, 15, 16*	⊢
	: , : , :
10	instantiation	17, 50, 95, 47, 18	,  ⊢
	: , : , :
11	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less
12	instantiation	19, 20, 21, 22*	,  ⊢
	: , : , :
13	instantiation	23, 173, 215, 24, 174, 25, 199	⊢
	: , : , : , : , :
14	instantiation	198, 44	⊢
	:
15	instantiation	68, 70, 44, 26	⊢
	: , :
16	instantiation	27, 97, 28, 144, 29	⊢
	: , : , :
17	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
18	instantiation	30, 95, 110, 67	⊢
	: , : , :
19	theorem		⊢
	proveit.logic.equality.sub_right_side_into
20	instantiation	31, 32, 33	,  ⊢
	: , : , :
21	instantiation	34, 201, 35, 36, 144, 86, 121, 37*	⊢
	: , : , :
22	instantiation	160, 38, 39	⊢
	: , : , :
23	theorem		⊢
	proveit.numbers.addition.term_as_strong_upper_bound
24	instantiation	109, 40	⊢
	:
25	instantiation	41, 42	⊢
	:
26	instantiation	90, 120, 92, 103, 43, 94	⊢
	: , : , :
27	theorem		⊢
	proveit.numbers.addition.subtraction.negated_add
28	instantiation	213, 189, 44	⊢
	: , : , :
29	instantiation	160, 45, 80	⊢
	: , : , :
30	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
31	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
32	instantiation	46, 50, 47, 110, 48	,  ⊢
	: , : , :
33	instantiation	49, 110, 50, 51, 52	⊢
	: , : , :
34	theorem		⊢
	proveit.numbers.division.distribute_frac_through_sum
35	instantiation	187	⊢
	: , :
36	instantiation	213, 189, 53	⊢
	: , : , :
37	instantiation	54, 85, 86, 121, 64*	⊢
	: , :
38	instantiation	146, 55	⊢
	: , : , :
39	instantiation	160, 56, 57	⊢
	: , : , :
40	instantiation	58, 215, 173, 174, 59	⊢
	: , : , : , : , :
41	theorem		⊢
	proveit.numbers.negation.real_neg_closure
42	instantiation	60, 61, 62, 121	⊢
	: , :
43	instantiation	63, 153, 184, 136	⊢
	: , : , :
44	instantiation	119, 103, 120, 121	⊢
	: , :
45	instantiation	146, 64	⊢
	: , : , :
46	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
47	instantiation	65, 95, 110, 67	⊢
	: , : , :
48	instantiation	66, 95, 110, 67	⊢
	: , : , :
49	theorem		⊢
	proveit.numbers.addition.weak_bound_via_right_term_bound
50	instantiation	198, 70	⊢
	:
51	instantiation	198, 69	⊢
	:
52	instantiation	68, 69, 70, 71	⊢
	: , :
53	instantiation	213, 206, 72	⊢
	: , : , :
54	theorem		⊢
	proveit.numbers.division.neg_frac_neg_numerator
55	instantiation	73, 74, 99, 75*	⊢
	: , :
56	instantiation	172, 215, 208, 173, 76, 174, 97, 79	⊢
	: , : , : , : , : , :
57	instantiation	77, 173, 208, 215, 174, 78, 97, 79, 80*	⊢
	: , : , : , : , : , :
58	theorem		⊢
	proveit.numbers.addition.add_nat_pos_from_nonneg
59	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
60	theorem		⊢
	proveit.numbers.division.div_real_pos_closure
61	instantiation	213, 82, 81	⊢
	: , : , :
62	instantiation	213, 82, 83	⊢
	: , : , :
63	theorem		⊢
	proveit.numbers.number_sets.integers.interval_upper_bound
64	instantiation	84, 85, 86, 121, 87*	⊢
	: , :
65	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
66	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_upper_bound
67	instantiation	88, 89	⊢
	:
68	theorem		⊢
	proveit.numbers.negation.negated_weak_bound
69	instantiation	119, 91, 120, 121	⊢
	: , :
70	instantiation	119, 92, 120, 121	⊢
	: , :
71	instantiation	90, 120, 91, 92, 93, 94	⊢
	: , : , :
72	instantiation	213, 211, 167	⊢
	: , : , :
73	theorem		⊢
	proveit.numbers.negation.distribute_neg_through_binary_sum
74	instantiation	213, 189, 95	⊢
	: , : , :
75	instantiation	96, 97	⊢
	:
76	instantiation	187	⊢
	: , :
77	theorem		⊢
	proveit.numbers.addition.association
78	instantiation	187	⊢
	: , :
79	instantiation	98, 99	⊢
	:
80	instantiation	100, 212, 202, 101*	⊢
	: , : , : , :
81	instantiation	213, 102, 150	⊢
	: , : , :
82	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
83	instantiation	213, 102, 137	⊢
	: , : , :
84	theorem		⊢
	proveit.numbers.division.div_as_mult
85	instantiation	213, 189, 103	⊢
	: , : , :
86	instantiation	213, 189, 120	⊢
	: , : , :
87	instantiation	160, 104, 105	⊢
	: , : , :
88	theorem		⊢
	proveit.physics.quantum.QPE._delta_b_in_interval
89	assumption		⊢
90	theorem		⊢
	proveit.numbers.division.weak_div_from_numer_bound__pos_denom
91	instantiation	213, 206, 106	⊢
	: , : , :
92	instantiation	213, 206, 107	⊢
	: , : , :
93	instantiation	108, 153, 184, 136	⊢
	: , : , :
94	instantiation	109, 137	⊢
	:
95	instantiation	198, 110	⊢
	:
96	theorem		⊢
	proveit.numbers.negation.double_negation
97	instantiation	213, 189, 110	⊢
	: , : , :
98	theorem		⊢
	proveit.numbers.negation.complex_closure
99	instantiation	213, 189, 111	⊢
	: , : , :
100	theorem		⊢
	proveit.numbers.addition.rational_pair_addition
101	instantiation	160, 112, 113	⊢
	: , : , :
102	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
103	instantiation	203, 204, 195	⊢
	: , : , :
104	instantiation	146, 114	⊢
	: , : , :
105	instantiation	115, 170, 116, 188, 130, 117*	⊢
	: , : , :
106	instantiation	213, 211, 153	⊢
	: , : , :
107	instantiation	213, 211, 118	⊢
	: , : , :
108	theorem		⊢
	proveit.numbers.number_sets.integers.interval_lower_bound
109	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
110	instantiation	119, 199, 185, 130	⊢
	: , :
111	instantiation	119, 199, 120, 121	⊢
	: , :
112	instantiation	154, 208, 122, 123, 124, 125	⊢
	: , : , : , :
113	instantiation	126, 127, 128	⊢
	:
114	instantiation	129, 170, 197, 190, 130, 131*	⊢
	: , : , :
115	theorem		⊢
	proveit.numbers.exponentiation.product_of_real_powers
116	instantiation	132, 197, 190	⊢
	: , :
117	instantiation	160, 133, 134	⊢
	: , : , :
118	instantiation	213, 135, 136	⊢
	: , : , :
119	theorem		⊢
	proveit.numbers.division.div_real_closure
120	instantiation	203, 204, 137	⊢
	: , : , :
121	instantiation	142, 137	⊢
	:
122	instantiation	187	⊢
	: , :
123	instantiation	187	⊢
	: , :
124	instantiation	160, 138, 139	⊢
	: , : , :
125	theorem		⊢
	proveit.numbers.numerals.decimals.mult_2_2
126	theorem		⊢
	proveit.numbers.division.frac_cancel_complete
127	instantiation	213, 189, 140	⊢
	: , : , :
128	instantiation	142, 141	⊢
	:
129	theorem		⊢
	proveit.numbers.exponentiation.real_power_of_real_power
130	instantiation	142, 201	⊢
	:
131	instantiation	143, 178, 144, 145*	⊢
	: , :
132	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
133	instantiation	146, 147	⊢
	: , : , :
134	instantiation	148, 149, 150, 151*	⊢
	: , :
135	instantiation	152, 153, 184	⊢
	: , :
136	assumption		⊢
137	theorem		⊢
	proveit.physics.quantum.QPE._two_pow_t_is_nat_pos
138	instantiation	154, 208, 155, 156, 157, 158	⊢
	: , : , : , :
139	theorem		⊢
	proveit.numbers.numerals.decimals.add_2_2
140	instantiation	213, 206, 159	⊢
	: , : , :
141	theorem		⊢
	proveit.numbers.numerals.decimals.posnat4
142	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
143	theorem		⊢
	proveit.numbers.multiplication.mult_neg_right
144	instantiation	213, 189, 199	⊢
	: , : , :
145	instantiation	169, 178	⊢
	:
146	axiom		⊢
	proveit.logic.equality.substitution
147	instantiation	160, 161, 162	⊢
	: , : , :
148	theorem		⊢
	proveit.numbers.exponentiation.neg_power_as_div
149	instantiation	213, 163, 164	⊢
	: , : , :
150	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
151	instantiation	165, 170	⊢
	:
152	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
153	instantiation	166, 167, 212	⊢
	: , :
154	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
155	instantiation	187	⊢
	: , :
156	instantiation	187	⊢
	: , :
157	instantiation	168, 170	⊢
	:
158	instantiation	169, 170	⊢
	:
159	instantiation	213, 211, 171	⊢
	: , : , :
160	axiom		⊢
	proveit.logic.equality.equals_transitivity
161	instantiation	172, 173, 208, 215, 174, 175, 178, 179, 176	⊢
	: , : , : , : , : , :
162	instantiation	177, 178, 179, 180	⊢
	: , : , :
163	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
164	instantiation	213, 181, 182	⊢
	: , : , :
165	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
166	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
167	instantiation	183, 184	⊢
	:
168	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
169	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
170	instantiation	213, 189, 185	⊢
	: , : , :
171	instantiation	213, 214, 186	⊢
	: , : , :
172	theorem		⊢
	proveit.numbers.addition.disassociation
173	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
174	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
175	instantiation	187	⊢
	: , :
176	instantiation	213, 189, 188	⊢
	: , : , :
177	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
178	instantiation	213, 189, 197	⊢
	: , : , :
179	instantiation	213, 189, 190	⊢
	: , : , :
180	instantiation	191	⊢
	:
181	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
182	instantiation	213, 192, 193	⊢
	: , : , :
183	theorem		⊢
	proveit.numbers.negation.int_closure
184	instantiation	213, 194, 195	⊢
	: , : , :
185	instantiation	213, 206, 196	⊢
	: , : , :
186	theorem		⊢
	proveit.numbers.numerals.decimals.nat4
187	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
188	instantiation	198, 197	⊢
	:
189	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
190	instantiation	198, 199	⊢
	:
191	axiom		⊢
	proveit.logic.equality.equals_reflexivity
192	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
193	instantiation	213, 200, 201	⊢
	: , : , :
194	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
195	theorem		⊢
	proveit.physics.quantum.QPE._two_pow_t_minus_one_is_nat_pos
196	instantiation	213, 211, 202	⊢
	: , : , :
197	instantiation	203, 204, 205	⊢
	: , : , :
198	theorem		⊢
	proveit.numbers.negation.real_closure
199	instantiation	213, 206, 207	⊢
	: , : , :
200	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
201	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
202	instantiation	213, 214, 208	⊢
	: , : , :
203	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
204	instantiation	209, 210	⊢
	: , :
205	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
206	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
207	instantiation	213, 211, 212	⊢
	: , : , :
208	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
209	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
210	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
211	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
212	instantiation	213, 214, 215	⊢
	: , : , :
213	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
214	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
215	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements