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, 7, 8, 9	⊢
	: , : , : , : , : , :
1	reference	94	⊢
2	reference	95	⊢
3	reference	193	⊢
4	reference	222	⊢
5	reference	96	⊢
6	instantiation	173	⊢
	: , :
7	instantiation	220, 182, 10	⊢
	: , : , :
8	instantiation	67, 175	⊢
	:
9	reference	175	⊢
10	instantiation	11, 167, 12	⊢
	: , :
11	theorem		⊢
	proveit.numbers.exponentiation.exp_real_closure_nat_power
12	instantiation	13, 14, 15	⊢
	:
13	theorem		⊢
	proveit.numbers.number_sets.integers.nonneg_int_is_natural
14	instantiation	16, 17, 18	⊢
	: , :
15	instantiation	19, 20	⊢
	: , :
16	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
17	instantiation	220, 21, 103	⊢
	: , : , :
18	instantiation	220, 22, 23	⊢
	: , : , :
19	theorem		⊢
	proveit.numbers.addition.subtraction.nonneg_difference
20	instantiation	61, 65, 167, 24, 25, 26*, 27*	⊢
	: , : , :
21	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
22	theorem		⊢
	proveit.numbers.number_sets.integers.neg_int_within_int
23	instantiation	28, 159	⊢
	:
24	instantiation	80, 30, 167	⊢
	: , :
25	instantiation	29, 167, 30, 31, 137	⊢
	: , : , :
26	instantiation	148, 32, 33	⊢
	: , : , :
27	instantiation	148, 34, 35	⊢
	: , : , :
28	theorem		⊢
	proveit.numbers.negation.int_neg_closure
29	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right
30	instantiation	80, 82, 120	⊢
	: , :
31	instantiation	46, 36, 37	⊢
	: , : , :
32	instantiation	94, 222, 193, 95, 52, 96, 154, 100, 53	⊢
	: , : , : , : , : , :
33	instantiation	99, 154, 100, 70	⊢
	: , : , :
34	instantiation	127, 38	⊢
	: , : , :
35	instantiation	39, 40, 41, 42	⊢
	: , : , : , :
36	instantiation	43, 219, 59, 44, 45*	⊢
	: , :
37	instantiation	46, 47, 48	⊢
	: , : , :
38	instantiation	94, 95, 193, 222, 96, 69, 72, 98, 154	⊢
	: , : , : , : , : , :
39	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
40	instantiation	94, 95, 50, 222, 96, 51, 72, 98, 154, 49	⊢
	: , : , : , : , : , :
41	instantiation	94, 50, 193, 95, 51, 52, 96, 72, 98, 154, 100, 53	⊢
	: , : , : , : , : , :
42	instantiation	148, 54, 55	⊢
	: , : , :
43	theorem		⊢
	proveit.numbers.rounding.ceil_of_real_above_int
44	instantiation	56, 202, 76, 57	⊢
	: , :
45	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
46	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
47	instantiation	58, 59, 171, 60	⊢
	: , :
48	instantiation	61, 120, 62, 82, 63, 64*	⊢
	: , : , :
49	instantiation	220, 182, 65	⊢
	: , : , :
50	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
51	instantiation	66	⊢
	: , : , :
52	instantiation	173	⊢
	: , :
53	instantiation	67, 154	⊢
	:
54	instantiation	68, 193, 222, 95, 69, 96, 72, 98, 154, 100, 70	⊢
	: , : , : , : , : , : , : , :
55	instantiation	71, 100, 72, 102	⊢
	: , : , :
56	theorem		⊢
	proveit.numbers.logarithms.log_base_large_a_greater_one
57	instantiation	73, 194, 74	⊢
	: , :
58	theorem		⊢
	proveit.numbers.rounding.ceil_increasing_less_eq
59	instantiation	177, 202, 76, 179	⊢
	: , :
60	instantiation	75, 202, 76, 178, 77, 194	⊢
	: , : , :
61	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
62	instantiation	80, 143, 121	⊢
	: , :
63	axiom		⊢
	proveit.physics.quantum.QPE._t_req
64	instantiation	148, 78, 79	⊢
	: , : , :
65	instantiation	80, 143, 81	⊢
	: , :
66	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
67	theorem		⊢
	proveit.numbers.negation.complex_closure
68	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_general
69	instantiation	173	⊢
	: , :
70	instantiation	122	⊢
	:
71	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
72	instantiation	220, 182, 82	⊢
	: , : , :
73	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
74	instantiation	83, 167, 84, 85, 86, 87*, 88*	⊢
	: , : , :
75	theorem		⊢
	proveit.numbers.logarithms.log_increasing_less_eq
76	instantiation	220, 204, 89	⊢
	: , : , :
77	instantiation	90, 167, 161, 91, 92, 93*	⊢
	: , : , :
78	instantiation	94, 95, 193, 222, 96, 97, 100, 101, 98	⊢
	: , : , : , : , : , :
79	instantiation	99, 100, 101, 102	⊢
	: , : , :
80	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
81	instantiation	142, 167	⊢
	:
82	instantiation	157, 158, 103	⊢
	: , : , :
83	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
84	instantiation	104, 161, 213	⊢
	: , :
85	instantiation	220, 216, 105	⊢
	: , : , :
86	instantiation	106, 161, 213, 214, 107, 108	⊢
	: , : , :
87	instantiation	148, 109, 110	⊢
	: , : , :
88	instantiation	148, 111, 112	⊢
	: , : , :
89	instantiation	188, 113, 205	⊢
	: , :
90	theorem		⊢
	proveit.numbers.addition.weak_bound_via_right_term_bound
91	instantiation	220, 114, 185	⊢
	: , : , :
92	instantiation	115, 116, 200, 202, 117	⊢
	: , : , :
93	instantiation	129, 183, 219, 130*, 118*, 119*	⊢
	: , : , : , :
94	theorem		⊢
	proveit.numbers.addition.disassociation
95	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
96	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
97	instantiation	173	⊢
	: , :
98	instantiation	220, 182, 120	⊢
	: , : , :
99	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
100	instantiation	220, 182, 143	⊢
	: , : , :
101	instantiation	220, 182, 121	⊢
	: , : , :
102	instantiation	122	⊢
	:
103	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
104	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
105	instantiation	123, 172, 217	⊢
	: , :
106	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_right_factor_bound
107	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
108	instantiation	124, 181	⊢
	:
109	instantiation	127, 125	⊢
	: , : , :
110	instantiation	126, 154	⊢
	:
111	instantiation	127, 128	⊢
	: , : , :
112	instantiation	129, 219, 183, 130*, 131*, 138*	⊢
	: , : , : , :
113	instantiation	220, 209, 132	⊢
	: , : , :
114	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
115	theorem		⊢
	proveit.numbers.division.weak_div_from_denom_bound__all_pos
116	instantiation	220, 133, 134	⊢
	: , : , :
117	instantiation	135, 167, 207, 214, 136, 137, 138*	⊢
	: , : , :
118	instantiation	148, 139, 140	⊢
	: , : , :
119	instantiation	141, 154	⊢
	:
120	instantiation	142, 143	⊢
	:
121	instantiation	220, 216, 144	⊢
	: , : , :
122	axiom		⊢
	proveit.logic.equality.equals_reflexivity
123	theorem		⊢
	proveit.numbers.multiplication.mult_rational_closure_bin
124	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
125	instantiation	145, 146	⊢
	:
126	theorem		⊢
	proveit.numbers.addition.elim_zero_left
127	axiom		⊢
	proveit.logic.equality.substitution
128	instantiation	174, 146	⊢
	:
129	theorem		⊢
	proveit.numbers.addition.rational_pair_addition
130	instantiation	147, 154	⊢
	:
131	instantiation	148, 149, 150	⊢
	: , : , :
132	theorem		⊢
	proveit.numbers.numerals.decimals.posnat5
133	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
134	instantiation	220, 151, 222	⊢
	: , : , :
135	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
136	instantiation	152, 213, 214, 215	⊢
	: , : , :
137	instantiation	153, 193	⊢
	:
138	instantiation	174, 154	⊢
	:
139	instantiation	162, 193, 155, 156, 166, 165	⊢
	: , : , : , :
140	theorem		⊢
	proveit.numbers.numerals.decimals.add_4_1
141	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
142	theorem		⊢
	proveit.numbers.negation.real_closure
143	instantiation	157, 158, 159	⊢
	: , : , :
144	instantiation	220, 218, 160	⊢
	: , : , :
145	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
146	instantiation	220, 182, 161	⊢
	: , : , :
147	theorem		⊢
	proveit.numbers.division.frac_one_denom
148	axiom		⊢
	proveit.logic.equality.equals_transitivity
149	instantiation	162, 193, 163, 164, 165, 166	⊢
	: , : , : , :
150	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_4
151	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
152	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_upper_bound
153	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_lower_bound
154	instantiation	220, 182, 167	⊢
	: , : , :
155	instantiation	173	⊢
	: , :
156	instantiation	173	⊢
	: , :
157	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
158	instantiation	168, 169	⊢
	: , :
159	axiom		⊢
	proveit.physics.quantum.QPE._n_in_natural_pos
160	instantiation	170, 171	⊢
	:
161	instantiation	220, 216, 172	⊢
	: , : , :
162	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
163	instantiation	173	⊢
	: , :
164	instantiation	173	⊢
	: , :
165	instantiation	174, 175	⊢
	:
166	theorem		⊢
	proveit.numbers.numerals.decimals.mult_2_2
167	instantiation	220, 216, 176	⊢
	: , : , :
168	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
169	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
170	axiom		⊢
	proveit.numbers.rounding.ceil_is_an_int
171	instantiation	177, 202, 178, 179	⊢
	: , :
172	instantiation	220, 180, 181	⊢
	: , : , :
173	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
174	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
175	instantiation	220, 182, 214	⊢
	: , : , :
176	instantiation	220, 218, 183	⊢
	: , : , :
177	theorem		⊢
	proveit.numbers.logarithms.log_real_pos_real_closure
178	instantiation	184, 202, 185	⊢
	: , :
179	instantiation	186, 187	⊢
	: , :
180	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
181	instantiation	188, 195, 205	⊢
	: , :
182	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
183	instantiation	220, 221, 193	⊢
	: , : , :
184	theorem		⊢
	proveit.numbers.addition.add_real_pos_closure_bin
185	instantiation	189, 190, 200, 191	⊢
	: , :
186	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
187	instantiation	192, 222, 193, 194	⊢
	: , :
188	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
189	theorem		⊢
	proveit.numbers.division.div_real_pos_closure
190	instantiation	220, 204, 195	⊢
	: , : , :
191	instantiation	196, 197	⊢
	:
192	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq_nat
193	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
194	theorem		⊢
	proveit.numbers.numerals.decimals.less_1_2
195	instantiation	220, 209, 198	⊢
	: , : , :
196	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
197	instantiation	220, 199, 200	⊢
	: , : , :
198	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
199	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
200	instantiation	201, 202, 203	⊢
	: , :
201	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
202	instantiation	220, 204, 205	⊢
	: , : , :
203	instantiation	206, 207, 208	⊢
	:
204	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
205	instantiation	220, 209, 210	⊢
	: , : , :
206	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pos_real_is_real_pos
207	instantiation	211, 213, 214, 215	⊢
	: , : , :
208	instantiation	212, 213, 214, 215	⊢
	: , : , :
209	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
210	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
211	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
212	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
213	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
214	instantiation	220, 216, 217	⊢
	: , : , :
215	axiom		⊢
	proveit.physics.quantum.QPE._eps_in_interval
216	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
217	instantiation	220, 218, 219	⊢
	: , : , :
218	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
219	instantiation	220, 221, 222	⊢
	: , : , :
220	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
221	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
222	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements