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.multiplication.weak_bound_via_right_factor_bound
2	instantiation	208, 198, 7	⊢
	: , : , :
3	instantiation	8, 10, 207	⊢
	: , :
4	instantiation	8, 11, 207	⊢
	: , :
5	instantiation	9, 180, 10, 11, 12, 13	⊢
	: , : , :
6	instantiation	14, 21	⊢
	:
7	instantiation	208, 205, 15	⊢
	: , : , :
8	theorem		⊢
	proveit.numbers.exponentiation.exp_real_closure_nat_power
9	theorem		⊢
	proveit.numbers.exponentiation.exp_pos_lesseq
10	instantiation	69, 189, 75	⊢
	: , :
11	instantiation	208, 198, 16	⊢
	: , : , :
12	instantiation	17, 18, 19	⊢
	: , :
13	instantiation	20, 204	⊢
	:
14	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_lower_bound
15	instantiation	208, 209, 21	⊢
	: , : , :
16	instantiation	22, 33, 23	⊢
	: , :
17	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
18	instantiation	24, 25	⊢
	:
19	instantiation	55, 173, 26, 27, 28, 29*	⊢
	: , : , :
20	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
21	theorem		⊢
	proveit.numbers.numerals.decimals.nat4
22	theorem		⊢
	proveit.numbers.addition.add_rational_closure_bin
23	instantiation	208, 205, 30	⊢
	: , : , :
24	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonneg_if_in_real_nonneg
25	instantiation	208, 31, 32	⊢
	: , : , :
26	instantiation	208, 171, 98	⊢
	: , : , :
27	instantiation	208, 198, 33	⊢
	: , : , :
28	instantiation	34, 180, 90, 35, 153, 36, 37*	⊢
	: , : , :
29	instantiation	133, 38, 39	⊢
	: , : , :
30	instantiation	208, 40, 41	⊢
	: , : , :
31	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonneg
32	instantiation	136, 182, 104	⊢
	: , :
33	instantiation	208, 42, 43	⊢
	: , : , :
34	theorem		⊢
	proveit.numbers.exponentiation.exp_monotonicity_large_base_less_eq
35	instantiation	69, 58, 56	⊢
	: , :
36	instantiation	44, 45, 46	⊢
	: , : , :
37	instantiation	47, 183, 98, 128	⊢
	: , :
38	instantiation	92, 145, 207, 210, 146, 48, 170, 64, 163	⊢
	: , : , : , : , : , :
39	instantiation	133, 49, 50	⊢
	: , : , :
40	theorem		⊢
	proveit.numbers.number_sets.integers.neg_int_within_int
41	instantiation	51, 202	⊢
	:
42	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_nonzero_within_rational
43	instantiation	52, 178, 53	⊢
	: , :
44	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
45	instantiation	54, 90	⊢
	:
46	instantiation	55, 56, 57, 58, 59, 60*	⊢
	: , : , :
47	theorem		⊢
	proveit.numbers.exponentiation.exponent_log_with_same_base
48	instantiation	159	⊢
	: , :
49	instantiation	61, 210, 145, 146, 170, 64, 163	⊢
	: , : , : , : , : , : , :
50	instantiation	62, 145, 207, 210, 146, 63, 170, 163, 64, 65*	⊢
	: , : , : , : , : , :
51	theorem		⊢
	proveit.numbers.negation.int_neg_closure
52	theorem		⊢
	proveit.numbers.exponentiation.exp_rational_nonzero_closure
53	instantiation	66, 67, 68	⊢
	: , :
54	theorem		⊢
	proveit.numbers.rounding.ceil_x_ge_x
55	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
56	instantiation	185, 106	⊢
	:
57	instantiation	69, 106, 70	⊢
	: , :
58	instantiation	109, 110, 78	⊢
	: , : , :
59	axiom		⊢
	proveit.physics.quantum.QPE._t_req
60	instantiation	71, 72, 73, 74	⊢
	: , : , : , :
61	theorem		⊢
	proveit.numbers.addition.leftward_commutation
62	theorem		⊢
	proveit.numbers.addition.association
63	instantiation	159	⊢
	: , :
64	instantiation	208, 179, 75	⊢
	: , : , :
65	instantiation	76, 143, 170, 77	⊢
	: , : , :
66	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
67	instantiation	208, 85, 78	⊢
	: , : , :
68	instantiation	79, 80	⊢
	:
69	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
70	instantiation	208, 198, 81	⊢
	: , : , :
71	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
72	instantiation	133, 82, 83	⊢
	: , : , :
73	instantiation	103	⊢
	:
74	instantiation	84, 91	⊢
	: , :
75	instantiation	208, 171, 104	⊢
	: , : , :
76	theorem		⊢
	proveit.numbers.addition.subtraction.subtract_from_add
77	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
78	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
79	theorem		⊢
	proveit.numbers.negation.int_closure
80	instantiation	208, 85, 111	⊢
	: , : , :
81	instantiation	208, 205, 86	⊢
	: , : , :
82	instantiation	131, 87	⊢
	: , : , :
83	instantiation	133, 88, 89	⊢
	: , : , :
84	theorem		⊢
	proveit.logic.equality.equals_reversal
85	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
86	instantiation	119, 90	⊢
	:
87	instantiation	131, 91	⊢
	: , : , :
88	instantiation	92, 145, 207, 210, 146, 93, 101, 96, 94	⊢
	: , : , : , : , : , :
89	instantiation	95, 101, 96, 97	⊢
	: , : , :
90	instantiation	126, 183, 98, 128	⊢
	: , :
91	instantiation	131, 99	⊢
	: , : , :
92	theorem		⊢
	proveit.numbers.addition.disassociation
93	instantiation	159	⊢
	: , :
94	instantiation	100, 101	⊢
	:
95	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
96	instantiation	208, 179, 102	⊢
	: , : , :
97	instantiation	103	⊢
	:
98	instantiation	136, 183, 104	⊢
	: , :
99	instantiation	131, 105	⊢
	: , : , :
100	theorem		⊢
	proveit.numbers.negation.complex_closure
101	instantiation	208, 179, 106	⊢
	: , : , :
102	instantiation	208, 198, 107	⊢
	: , : , :
103	axiom		⊢
	proveit.logic.equality.equals_reflexivity
104	instantiation	181, 182, 130, 115	⊢
	: , :
105	instantiation	131, 108	⊢
	: , : , :
106	instantiation	109, 110, 111	⊢
	: , : , :
107	instantiation	208, 205, 112	⊢
	: , : , :
108	instantiation	113, 143, 114, 115, 116*	⊢
	: , :
109	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
110	instantiation	117, 118	⊢
	: , :
111	axiom		⊢
	proveit.physics.quantum.QPE._n_in_natural_pos
112	instantiation	119, 120	⊢
	:
113	theorem		⊢
	proveit.numbers.division.div_as_mult
114	instantiation	208, 179, 121	⊢
	: , : , :
115	instantiation	122, 123	⊢
	:
116	instantiation	133, 124, 125	⊢
	: , : , :
117	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
118	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
119	axiom		⊢
	proveit.numbers.rounding.ceil_is_an_int
120	instantiation	126, 183, 127, 128	⊢
	: , :
121	instantiation	208, 171, 130	⊢
	: , : , :
122	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
123	instantiation	208, 129, 130	⊢
	: , : , :
124	instantiation	131, 132	⊢
	: , : , :
125	instantiation	133, 134, 135	⊢
	: , : , :
126	theorem		⊢
	proveit.numbers.logarithms.log_real_pos_real_closure
127	instantiation	136, 183, 137	⊢
	: , :
128	instantiation	154, 138	⊢
	: , :
129	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
130	instantiation	150, 183, 165	⊢
	: , :
131	axiom		⊢
	proveit.logic.equality.substitution
132	instantiation	139, 170, 162, 173, 184, 140, 141*	⊢
	: , : , :
133	axiom		⊢
	proveit.logic.equality.equals_transitivity
134	instantiation	142, 210, 207, 145, 147, 146, 143, 148, 149	⊢
	: , : , : , : , : , :
135	instantiation	144, 145, 207, 146, 147, 148, 149	⊢
	: , : , : , :
136	theorem		⊢
	proveit.numbers.addition.add_real_pos_closure_bin
137	instantiation	150, 172, 151	⊢
	: , :
138	instantiation	152, 210, 207, 153	⊢
	: , :
139	theorem		⊢
	proveit.numbers.exponentiation.real_power_of_product
140	instantiation	154, 155	⊢
	: , :
141	instantiation	156, 157, 202, 158*	⊢
	: , :
142	theorem		⊢
	proveit.numbers.multiplication.disassociation
143	instantiation	208, 179, 189	⊢
	: , : , :
144	theorem		⊢
	proveit.numbers.multiplication.elim_one_any
145	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
146	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
147	instantiation	159	⊢
	: , :
148	instantiation	208, 179, 160	⊢
	: , : , :
149	instantiation	161, 162, 163	⊢
	: , :
150	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
151	instantiation	164, 165, 173	⊢
	: , :
152	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq_nat
153	theorem		⊢
	proveit.numbers.numerals.decimals.less_1_2
154	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
155	instantiation	166, 188, 175, 176	⊢
	: , :
156	theorem		⊢
	proveit.numbers.exponentiation.neg_power_as_div
157	instantiation	208, 167, 168	⊢
	: , : , :
158	instantiation	169, 170	⊢
	:
159	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
160	instantiation	208, 171, 172	⊢
	: , : , :
161	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
162	instantiation	208, 179, 175	⊢
	: , : , :
163	instantiation	208, 179, 173	⊢
	: , : , :
164	theorem		⊢
	proveit.numbers.exponentiation.exp_real_pos_closure
165	instantiation	174, 175, 176	⊢
	:
166	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq
167	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
168	instantiation	208, 177, 178	⊢
	: , : , :
169	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
170	instantiation	208, 179, 180	⊢
	: , : , :
171	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
172	instantiation	181, 182, 183, 184	⊢
	: , :
173	instantiation	185, 189	⊢
	:
174	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pos_real_is_real_pos
175	instantiation	186, 188, 189, 190	⊢
	: , : , :
176	instantiation	187, 188, 189, 190	⊢
	: , : , :
177	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
178	instantiation	208, 191, 192	⊢
	: , : , :
179	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
180	instantiation	208, 198, 193	⊢
	: , : , :
181	theorem		⊢
	proveit.numbers.division.div_real_pos_closure
182	instantiation	208, 195, 194	⊢
	: , : , :
183	instantiation	208, 195, 196	⊢
	: , : , :
184	instantiation	197, 204	⊢
	:
185	theorem		⊢
	proveit.numbers.negation.real_closure
186	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
187	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
188	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
189	instantiation	208, 198, 199	⊢
	: , : , :
190	axiom		⊢
	proveit.physics.quantum.QPE._eps_in_interval
191	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
192	instantiation	208, 200, 204	⊢
	: , : , :
193	instantiation	208, 205, 201	⊢
	: , : , :
194	instantiation	208, 203, 202	⊢
	: , : , :
195	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
196	instantiation	208, 203, 204	⊢
	: , : , :
197	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
198	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
199	instantiation	208, 205, 206	⊢
	: , : , :
200	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
201	instantiation	208, 209, 207	⊢
	: , : , :
202	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
203	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
204	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
205	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
206	instantiation	208, 209, 210	⊢
	: , : , :
207	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
208	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
209	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
210	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements