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	⊢
	:
1	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_lower_bound
2	instantiation	3, 187, 4	⊢
	: , :
3	theorem		⊢
	proveit.numbers.exponentiation.exp_natpos_closure
4	instantiation	5, 6, 7	⊢
	:
5	theorem		⊢
	proveit.numbers.number_sets.integers.nonneg_int_is_natural
6	instantiation	8, 9, 10	⊢
	: , :
7	instantiation	11, 12	⊢
	: , :
8	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
9	instantiation	214, 13, 97	⊢
	: , : , :
10	instantiation	214, 14, 15	⊢
	: , : , :
11	theorem		⊢
	proveit.numbers.addition.subtraction.nonneg_difference
12	instantiation	40, 16, 17	⊢
	: , : , :
13	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
14	theorem		⊢
	proveit.numbers.number_sets.integers.neg_int_within_int
15	instantiation	18, 204	⊢
	:
16	axiom		⊢
	proveit.physics.quantum.QPE._n_ge_two
17	instantiation	55, 59, 161, 19, 20, 21*, 22*	⊢
	: , : , :
18	theorem		⊢
	proveit.numbers.negation.int_neg_closure
19	instantiation	74, 24, 161	⊢
	: , :
20	instantiation	23, 161, 24, 25, 131	⊢
	: , : , :
21	instantiation	142, 26, 27	⊢
	: , : , :
22	instantiation	142, 28, 29	⊢
	: , : , :
23	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right
24	instantiation	74, 76, 114	⊢
	: , :
25	instantiation	40, 30, 31	⊢
	: , : , :
26	instantiation	88, 216, 187, 89, 46, 90, 148, 94, 47	⊢
	: , : , : , : , : , :
27	instantiation	93, 148, 94, 64	⊢
	: , : , :
28	instantiation	121, 32	⊢
	: , : , :
29	instantiation	33, 34, 35, 36	⊢
	: , : , : , :
30	instantiation	37, 213, 53, 38, 39*	⊢
	: , :
31	instantiation	40, 41, 42	⊢
	: , : , :
32	instantiation	88, 89, 187, 216, 90, 63, 66, 92, 148	⊢
	: , : , : , : , : , :
33	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
34	instantiation	88, 89, 44, 216, 90, 45, 66, 92, 148, 43	⊢
	: , : , : , : , : , :
35	instantiation	88, 44, 187, 89, 45, 46, 90, 66, 92, 148, 94, 47	⊢
	: , : , : , : , : , :
36	instantiation	142, 48, 49	⊢
	: , : , :
37	theorem		⊢
	proveit.numbers.rounding.ceil_of_real_above_int
38	instantiation	50, 196, 70, 51	⊢
	: , :
39	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
40	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
41	instantiation	52, 53, 165, 54	⊢
	: , :
42	instantiation	55, 114, 56, 76, 57, 58*	⊢
	: , : , :
43	instantiation	214, 176, 59	⊢
	: , : , :
44	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
45	instantiation	60	⊢
	: , : , :
46	instantiation	167	⊢
	: , :
47	instantiation	61, 148	⊢
	:
48	instantiation	62, 187, 216, 89, 63, 90, 66, 92, 148, 94, 64	⊢
	: , : , : , : , : , : , : , :
49	instantiation	65, 94, 66, 96	⊢
	: , : , :
50	theorem		⊢
	proveit.numbers.logarithms.log_base_large_a_greater_one
51	instantiation	67, 188, 68	⊢
	: , :
52	theorem		⊢
	proveit.numbers.rounding.ceil_increasing_less_eq
53	instantiation	171, 196, 70, 173	⊢
	: , :
54	instantiation	69, 196, 70, 172, 71, 188	⊢
	: , : , :
55	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
56	instantiation	74, 137, 115	⊢
	: , :
57	axiom		⊢
	proveit.physics.quantum.QPE._t_req
58	instantiation	142, 72, 73	⊢
	: , : , :
59	instantiation	74, 137, 75	⊢
	: , :
60	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
61	theorem		⊢
	proveit.numbers.negation.complex_closure
62	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_general
63	instantiation	167	⊢
	: , :
64	instantiation	116	⊢
	:
65	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
66	instantiation	214, 176, 76	⊢
	: , : , :
67	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
68	instantiation	77, 161, 78, 79, 80, 81*, 82*	⊢
	: , : , :
69	theorem		⊢
	proveit.numbers.logarithms.log_increasing_less_eq
70	instantiation	214, 198, 83	⊢
	: , : , :
71	instantiation	84, 161, 155, 85, 86, 87*	⊢
	: , : , :
72	instantiation	88, 89, 187, 216, 90, 91, 94, 95, 92	⊢
	: , : , : , : , : , :
73	instantiation	93, 94, 95, 96	⊢
	: , : , :
74	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
75	instantiation	136, 161	⊢
	:
76	instantiation	151, 152, 97	⊢
	: , : , :
77	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
78	instantiation	98, 155, 207	⊢
	: , :
79	instantiation	214, 210, 99	⊢
	: , : , :
80	instantiation	100, 155, 207, 208, 101, 102	⊢
	: , : , :
81	instantiation	142, 103, 104	⊢
	: , : , :
82	instantiation	142, 105, 106	⊢
	: , : , :
83	instantiation	182, 107, 199	⊢
	: , :
84	theorem		⊢
	proveit.numbers.addition.weak_bound_via_right_term_bound
85	instantiation	214, 108, 179	⊢
	: , : , :
86	instantiation	109, 110, 194, 196, 111	⊢
	: , : , :
87	instantiation	123, 177, 213, 124*, 112*, 113*	⊢
	: , : , : , :
88	theorem		⊢
	proveit.numbers.addition.disassociation
89	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
90	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
91	instantiation	167	⊢
	: , :
92	instantiation	214, 176, 114	⊢
	: , : , :
93	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
94	instantiation	214, 176, 137	⊢
	: , : , :
95	instantiation	214, 176, 115	⊢
	: , : , :
96	instantiation	116	⊢
	:
97	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
98	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
99	instantiation	117, 166, 211	⊢
	: , :
100	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_right_factor_bound
101	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
102	instantiation	118, 175	⊢
	:
103	instantiation	121, 119	⊢
	: , : , :
104	instantiation	120, 148	⊢
	:
105	instantiation	121, 122	⊢
	: , : , :
106	instantiation	123, 213, 177, 124*, 125*, 132*	⊢
	: , : , : , :
107	instantiation	214, 203, 126	⊢
	: , : , :
108	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
109	theorem		⊢
	proveit.numbers.division.weak_div_from_denom_bound__all_pos
110	instantiation	214, 127, 128	⊢
	: , : , :
111	instantiation	129, 161, 201, 208, 130, 131, 132*	⊢
	: , : , :
112	instantiation	142, 133, 134	⊢
	: , : , :
113	instantiation	135, 148	⊢
	:
114	instantiation	136, 137	⊢
	:
115	instantiation	214, 210, 138	⊢
	: , : , :
116	axiom		⊢
	proveit.logic.equality.equals_reflexivity
117	theorem		⊢
	proveit.numbers.multiplication.mult_rational_closure_bin
118	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
119	instantiation	139, 140	⊢
	:
120	theorem		⊢
	proveit.numbers.addition.elim_zero_left
121	axiom		⊢
	proveit.logic.equality.substitution
122	instantiation	168, 140	⊢
	:
123	theorem		⊢
	proveit.numbers.addition.rational_pair_addition
124	instantiation	141, 148	⊢
	:
125	instantiation	142, 143, 144	⊢
	: , : , :
126	theorem		⊢
	proveit.numbers.numerals.decimals.posnat5
127	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
128	instantiation	214, 145, 216	⊢
	: , : , :
129	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
130	instantiation	146, 207, 208, 209	⊢
	: , : , :
131	instantiation	147, 187	⊢
	:
132	instantiation	168, 148	⊢
	:
133	instantiation	156, 187, 149, 150, 160, 159	⊢
	: , : , : , :
134	theorem		⊢
	proveit.numbers.numerals.decimals.add_4_1
135	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
136	theorem		⊢
	proveit.numbers.negation.real_closure
137	instantiation	151, 152, 153	⊢
	: , : , :
138	instantiation	214, 212, 154	⊢
	: , : , :
139	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
140	instantiation	214, 176, 155	⊢
	: , : , :
141	theorem		⊢
	proveit.numbers.division.frac_one_denom
142	axiom		⊢
	proveit.logic.equality.equals_transitivity
143	instantiation	156, 187, 157, 158, 159, 160	⊢
	: , : , : , :
144	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_4
145	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
146	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_upper_bound
147	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_lower_bound
148	instantiation	214, 176, 161	⊢
	: , : , :
149	instantiation	167	⊢
	: , :
150	instantiation	167	⊢
	: , :
151	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
152	instantiation	162, 163	⊢
	: , :
153	axiom		⊢
	proveit.physics.quantum.QPE._n_in_natural_pos
154	instantiation	164, 165	⊢
	:
155	instantiation	214, 210, 166	⊢
	: , : , :
156	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
157	instantiation	167	⊢
	: , :
158	instantiation	167	⊢
	: , :
159	instantiation	168, 169	⊢
	:
160	theorem		⊢
	proveit.numbers.numerals.decimals.mult_2_2
161	instantiation	214, 210, 170	⊢
	: , : , :
162	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
163	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
164	axiom		⊢
	proveit.numbers.rounding.ceil_is_an_int
165	instantiation	171, 196, 172, 173	⊢
	: , :
166	instantiation	214, 174, 175	⊢
	: , : , :
167	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
168	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
169	instantiation	214, 176, 208	⊢
	: , : , :
170	instantiation	214, 212, 177	⊢
	: , : , :
171	theorem		⊢
	proveit.numbers.logarithms.log_real_pos_real_closure
172	instantiation	178, 196, 179	⊢
	: , :
173	instantiation	180, 181	⊢
	: , :
174	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
175	instantiation	182, 189, 199	⊢
	: , :
176	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
177	instantiation	214, 215, 187	⊢
	: , : , :
178	theorem		⊢
	proveit.numbers.addition.add_real_pos_closure_bin
179	instantiation	183, 184, 194, 185	⊢
	: , :
180	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
181	instantiation	186, 216, 187, 188	⊢
	: , :
182	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
183	theorem		⊢
	proveit.numbers.division.div_real_pos_closure
184	instantiation	214, 198, 189	⊢
	: , : , :
185	instantiation	190, 191	⊢
	:
186	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq_nat
187	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
188	theorem		⊢
	proveit.numbers.numerals.decimals.less_1_2
189	instantiation	214, 203, 192	⊢
	: , : , :
190	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
191	instantiation	214, 193, 194	⊢
	: , : , :
192	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
193	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
194	instantiation	195, 196, 197	⊢
	: , :
195	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
196	instantiation	214, 198, 199	⊢
	: , : , :
197	instantiation	200, 201, 202	⊢
	:
198	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
199	instantiation	214, 203, 204	⊢
	: , : , :
200	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pos_real_is_real_pos
201	instantiation	205, 207, 208, 209	⊢
	: , : , :
202	instantiation	206, 207, 208, 209	⊢
	: , : , :
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.real_numbers.all_in_interval_oc__is__real
206	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
207	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
208	instantiation	214, 210, 211	⊢
	: , : , :
209	axiom		⊢
	proveit.physics.quantum.QPE._eps_in_interval
210	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
211	instantiation	214, 212, 213	⊢
	: , : , :
212	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
213	instantiation	214, 215, 216	⊢
	: , : , :
214	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
215	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
216	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements