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	reference	126	⊢
2	instantiation	147, 3, 4	⊢
	: , : , :
3	instantiation	93, 94, 192, 221, 95, 5, 7, 6, 174	⊢
	: , : , : , : , : , :
4	instantiation	70, 174, 7, 8	⊢
	: , : , :
5	instantiation	172	⊢
	: , :
6	instantiation	66, 174	⊢
	:
7	instantiation	219, 181, 9	⊢
	: , : , :
8	instantiation	121	⊢
	:
9	instantiation	10, 166, 11	⊢
	: , :
10	theorem		⊢
	proveit.numbers.exponentiation.exp_real_closure_nat_power
11	instantiation	12, 13, 14	⊢
	:
12	theorem		⊢
	proveit.numbers.number_sets.integers.nonneg_int_is_natural
13	instantiation	15, 16, 17	⊢
	: , :
14	instantiation	18, 19	⊢
	: , :
15	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
16	instantiation	219, 20, 102	⊢
	: , : , :
17	instantiation	219, 21, 22	⊢
	: , : , :
18	theorem		⊢
	proveit.numbers.addition.subtraction.nonneg_difference
19	instantiation	60, 64, 166, 23, 24, 25*, 26*	⊢
	: , : , :
20	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
21	theorem		⊢
	proveit.numbers.number_sets.integers.neg_int_within_int
22	instantiation	27, 158	⊢
	:
23	instantiation	79, 29, 166	⊢
	: , :
24	instantiation	28, 166, 29, 30, 136	⊢
	: , : , :
25	instantiation	147, 31, 32	⊢
	: , : , :
26	instantiation	147, 33, 34	⊢
	: , : , :
27	theorem		⊢
	proveit.numbers.negation.int_neg_closure
28	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right
29	instantiation	79, 81, 119	⊢
	: , :
30	instantiation	45, 35, 36	⊢
	: , : , :
31	instantiation	93, 221, 192, 94, 51, 95, 153, 99, 52	⊢
	: , : , : , : , : , :
32	instantiation	98, 153, 99, 69	⊢
	: , : , :
33	instantiation	126, 37	⊢
	: , : , :
34	instantiation	38, 39, 40, 41	⊢
	: , : , : , :
35	instantiation	42, 218, 58, 43, 44*	⊢
	: , :
36	instantiation	45, 46, 47	⊢
	: , : , :
37	instantiation	93, 94, 192, 221, 95, 68, 71, 97, 153	⊢
	: , : , : , : , : , :
38	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
39	instantiation	93, 94, 49, 221, 95, 50, 71, 97, 153, 48	⊢
	: , : , : , : , : , :
40	instantiation	93, 49, 192, 94, 50, 51, 95, 71, 97, 153, 99, 52	⊢
	: , : , : , : , : , :
41	instantiation	147, 53, 54	⊢
	: , : , :
42	theorem		⊢
	proveit.numbers.rounding.ceil_of_real_above_int
43	instantiation	55, 201, 75, 56	⊢
	: , :
44	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
45	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
46	instantiation	57, 58, 170, 59	⊢
	: , :
47	instantiation	60, 119, 61, 81, 62, 63*	⊢
	: , : , :
48	instantiation	219, 181, 64	⊢
	: , : , :
49	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
50	instantiation	65	⊢
	: , : , :
51	instantiation	172	⊢
	: , :
52	instantiation	66, 153	⊢
	:
53	instantiation	67, 192, 221, 94, 68, 95, 71, 97, 153, 99, 69	⊢
	: , : , : , : , : , : , : , :
54	instantiation	70, 99, 71, 101	⊢
	: , : , :
55	theorem		⊢
	proveit.numbers.logarithms.log_base_large_a_greater_one
56	instantiation	72, 193, 73	⊢
	: , :
57	theorem		⊢
	proveit.numbers.rounding.ceil_increasing_less_eq
58	instantiation	176, 201, 75, 178	⊢
	: , :
59	instantiation	74, 201, 75, 177, 76, 193	⊢
	: , : , :
60	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
61	instantiation	79, 142, 120	⊢
	: , :
62	axiom		⊢
	proveit.physics.quantum.QPE._t_req
63	instantiation	147, 77, 78	⊢
	: , : , :
64	instantiation	79, 142, 80	⊢
	: , :
65	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
66	theorem		⊢
	proveit.numbers.negation.complex_closure
67	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_general
68	instantiation	172	⊢
	: , :
69	instantiation	121	⊢
	:
70	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
71	instantiation	219, 181, 81	⊢
	: , : , :
72	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
73	instantiation	82, 166, 83, 84, 85, 86*, 87*	⊢
	: , : , :
74	theorem		⊢
	proveit.numbers.logarithms.log_increasing_less_eq
75	instantiation	219, 203, 88	⊢
	: , : , :
76	instantiation	89, 166, 160, 90, 91, 92*	⊢
	: , : , :
77	instantiation	93, 94, 192, 221, 95, 96, 99, 100, 97	⊢
	: , : , : , : , : , :
78	instantiation	98, 99, 100, 101	⊢
	: , : , :
79	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
80	instantiation	141, 166	⊢
	:
81	instantiation	156, 157, 102	⊢
	: , : , :
82	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
83	instantiation	103, 160, 212	⊢
	: , :
84	instantiation	219, 215, 104	⊢
	: , : , :
85	instantiation	105, 160, 212, 213, 106, 107	⊢
	: , : , :
86	instantiation	147, 108, 109	⊢
	: , : , :
87	instantiation	147, 110, 111	⊢
	: , : , :
88	instantiation	187, 112, 204	⊢
	: , :
89	theorem		⊢
	proveit.numbers.addition.weak_bound_via_right_term_bound
90	instantiation	219, 113, 184	⊢
	: , : , :
91	instantiation	114, 115, 199, 201, 116	⊢
	: , : , :
92	instantiation	128, 182, 218, 129*, 117*, 118*	⊢
	: , : , : , :
93	theorem		⊢
	proveit.numbers.addition.disassociation
94	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
95	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
96	instantiation	172	⊢
	: , :
97	instantiation	219, 181, 119	⊢
	: , : , :
98	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
99	instantiation	219, 181, 142	⊢
	: , : , :
100	instantiation	219, 181, 120	⊢
	: , : , :
101	instantiation	121	⊢
	:
102	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
103	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
104	instantiation	122, 171, 216	⊢
	: , :
105	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_right_factor_bound
106	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
107	instantiation	123, 180	⊢
	:
108	instantiation	126, 124	⊢
	: , : , :
109	instantiation	125, 153	⊢
	:
110	instantiation	126, 127	⊢
	: , : , :
111	instantiation	128, 218, 182, 129*, 130*, 137*	⊢
	: , : , : , :
112	instantiation	219, 208, 131	⊢
	: , : , :
113	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
114	theorem		⊢
	proveit.numbers.division.weak_div_from_denom_bound__all_pos
115	instantiation	219, 132, 133	⊢
	: , : , :
116	instantiation	134, 166, 206, 213, 135, 136, 137*	⊢
	: , : , :
117	instantiation	147, 138, 139	⊢
	: , : , :
118	instantiation	140, 153	⊢
	:
119	instantiation	141, 142	⊢
	:
120	instantiation	219, 215, 143	⊢
	: , : , :
121	axiom		⊢
	proveit.logic.equality.equals_reflexivity
122	theorem		⊢
	proveit.numbers.multiplication.mult_rational_closure_bin
123	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
124	instantiation	144, 145	⊢
	:
125	theorem		⊢
	proveit.numbers.addition.elim_zero_left
126	axiom		⊢
	proveit.logic.equality.substitution
127	instantiation	173, 145	⊢
	:
128	theorem		⊢
	proveit.numbers.addition.rational_pair_addition
129	instantiation	146, 153	⊢
	:
130	instantiation	147, 148, 149	⊢
	: , : , :
131	theorem		⊢
	proveit.numbers.numerals.decimals.posnat5
132	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
133	instantiation	219, 150, 221	⊢
	: , : , :
134	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
135	instantiation	151, 212, 213, 214	⊢
	: , : , :
136	instantiation	152, 192	⊢
	:
137	instantiation	173, 153	⊢
	:
138	instantiation	161, 192, 154, 155, 165, 164	⊢
	: , : , : , :
139	theorem		⊢
	proveit.numbers.numerals.decimals.add_4_1
140	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
141	theorem		⊢
	proveit.numbers.negation.real_closure
142	instantiation	156, 157, 158	⊢
	: , : , :
143	instantiation	219, 217, 159	⊢
	: , : , :
144	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
145	instantiation	219, 181, 160	⊢
	: , : , :
146	theorem		⊢
	proveit.numbers.division.frac_one_denom
147	axiom		⊢
	proveit.logic.equality.equals_transitivity
148	instantiation	161, 192, 162, 163, 164, 165	⊢
	: , : , : , :
149	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_4
150	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
151	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_upper_bound
152	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_lower_bound
153	instantiation	219, 181, 166	⊢
	: , : , :
154	instantiation	172	⊢
	: , :
155	instantiation	172	⊢
	: , :
156	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
157	instantiation	167, 168	⊢
	: , :
158	axiom		⊢
	proveit.physics.quantum.QPE._n_in_natural_pos
159	instantiation	169, 170	⊢
	:
160	instantiation	219, 215, 171	⊢
	: , : , :
161	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
162	instantiation	172	⊢
	: , :
163	instantiation	172	⊢
	: , :
164	instantiation	173, 174	⊢
	:
165	theorem		⊢
	proveit.numbers.numerals.decimals.mult_2_2
166	instantiation	219, 215, 175	⊢
	: , : , :
167	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
168	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
169	axiom		⊢
	proveit.numbers.rounding.ceil_is_an_int
170	instantiation	176, 201, 177, 178	⊢
	: , :
171	instantiation	219, 179, 180	⊢
	: , : , :
172	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
173	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
174	instantiation	219, 181, 213	⊢
	: , : , :
175	instantiation	219, 217, 182	⊢
	: , : , :
176	theorem		⊢
	proveit.numbers.logarithms.log_real_pos_real_closure
177	instantiation	183, 201, 184	⊢
	: , :
178	instantiation	185, 186	⊢
	: , :
179	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
180	instantiation	187, 194, 204	⊢
	: , :
181	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
182	instantiation	219, 220, 192	⊢
	: , : , :
183	theorem		⊢
	proveit.numbers.addition.add_real_pos_closure_bin
184	instantiation	188, 189, 199, 190	⊢
	: , :
185	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
186	instantiation	191, 221, 192, 193	⊢
	: , :
187	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
188	theorem		⊢
	proveit.numbers.division.div_real_pos_closure
189	instantiation	219, 203, 194	⊢
	: , : , :
190	instantiation	195, 196	⊢
	:
191	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq_nat
192	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
193	theorem		⊢
	proveit.numbers.numerals.decimals.less_1_2
194	instantiation	219, 208, 197	⊢
	: , : , :
195	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
196	instantiation	219, 198, 199	⊢
	: , : , :
197	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
198	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
199	instantiation	200, 201, 202	⊢
	: , :
200	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
201	instantiation	219, 203, 204	⊢
	: , : , :
202	instantiation	205, 206, 207	⊢
	:
203	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
204	instantiation	219, 208, 209	⊢
	: , : , :
205	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pos_real_is_real_pos
206	instantiation	210, 212, 213, 214	⊢
	: , : , :
207	instantiation	211, 212, 213, 214	⊢
	: , : , :
208	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
209	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
210	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
211	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
212	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
213	instantiation	219, 215, 216	⊢
	: , : , :
214	axiom		⊢
	proveit.physics.quantum.QPE._eps_in_interval
215	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
216	instantiation	219, 217, 218	⊢
	: , : , :
217	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
218	instantiation	219, 220, 221	⊢
	: , : , :
219	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
220	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
221	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements