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	39	⊢
2	reference	124	⊢
3	reference	222	⊢
4	reference	125	⊢
5	instantiation	202	⊢
	: , :
6	instantiation	202	⊢
	: , :
7	reference	18	⊢
8	instantiation	96, 204	⊢
	:
9	instantiation	10, 11, 12*	⊢
	: , :
10	theorem		⊢
	proveit.logic.equality.equals_reversal
11	instantiation	13, 124, 222, 251, 125, 14, 204, 18, 15*	⊢
	: , : , : , : , : , :
12	instantiation	177, 16, 17	⊢
	: , : , :
13	theorem		⊢
	proveit.numbers.multiplication.distribute_through_sum
14	instantiation	202	⊢
	: , :
15	instantiation	170, 18	⊢
	:
16	instantiation	156, 74	⊢
	: , : , :
17	instantiation	19, 183, 243, 20, 21, 22*, 23*	⊢
	: , : , :
18	instantiation	249, 211, 24	⊢
	: , : , :
19	theorem		⊢
	proveit.numbers.exponentiation.product_of_real_powers
20	instantiation	249, 245, 25	⊢
	: , : , :
21	instantiation	26, 239	⊢
	:
22	instantiation	27, 183	⊢
	:
23	instantiation	177, 28, 29	⊢
	: , : , :
24	instantiation	30, 196, 31	⊢
	: , :
25	instantiation	249, 247, 36	⊢
	: , : , :
26	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
27	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
28	instantiation	123, 251, 222, 124, 32, 125, 204, 101, 82	⊢
	: , : , : , : , : , :
29	instantiation	177, 33, 34	⊢
	: , : , :
30	theorem		⊢
	proveit.numbers.exponentiation.exp_real_closure_nat_power
31	instantiation	35, 36, 37	⊢
	:
32	instantiation	202	⊢
	: , :
33	instantiation	38, 251, 124, 125, 204, 101, 82	⊢
	: , : , : , : , : , : , :
34	instantiation	39, 124, 222, 251, 125, 40, 204, 82, 101, 41*	⊢
	: , : , : , : , : , :
35	theorem		⊢
	proveit.numbers.number_sets.integers.nonneg_int_is_natural
36	instantiation	42, 43, 44	⊢
	: , :
37	instantiation	45, 46	⊢
	: , :
38	theorem		⊢
	proveit.numbers.addition.leftward_commutation
39	theorem		⊢
	proveit.numbers.addition.association
40	instantiation	202	⊢
	: , :
41	instantiation	47, 204, 183, 74	⊢
	: , : , :
42	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
43	instantiation	249, 48, 132	⊢
	: , : , :
44	instantiation	249, 49, 50	⊢
	: , : , :
45	theorem		⊢
	proveit.numbers.addition.subtraction.nonneg_difference
46	instantiation	75, 51, 52	⊢
	: , : , :
47	theorem		⊢
	proveit.numbers.addition.subtraction.subtract_from_add_reversed
48	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
49	theorem		⊢
	proveit.numbers.number_sets.integers.neg_int_within_int
50	instantiation	53, 239	⊢
	:
51	axiom		⊢
	proveit.physics.quantum.QPE._n_ge_two
52	instantiation	90, 94, 196, 54, 55, 56*, 57*	⊢
	: , : , :
53	theorem		⊢
	proveit.numbers.negation.int_neg_closure
54	instantiation	109, 59, 196	⊢
	: , :
55	instantiation	58, 196, 59, 60, 166	⊢
	: , : , :
56	instantiation	177, 61, 62	⊢
	: , : , :
57	instantiation	177, 63, 64	⊢
	: , : , :
58	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right
59	instantiation	109, 111, 149	⊢
	: , :
60	instantiation	75, 65, 66	⊢
	: , : , :
61	instantiation	123, 251, 222, 124, 81, 125, 183, 129, 82	⊢
	: , : , : , : , : , :
62	instantiation	128, 183, 129, 99	⊢
	: , : , :
63	instantiation	156, 67	⊢
	: , : , :
64	instantiation	68, 69, 70, 71	⊢
	: , : , : , :
65	instantiation	72, 248, 88, 73, 74*	⊢
	: , :
66	instantiation	75, 76, 77	⊢
	: , : , :
67	instantiation	123, 124, 222, 251, 125, 98, 101, 127, 183	⊢
	: , : , : , : , : , :
68	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
69	instantiation	123, 124, 79, 251, 125, 80, 101, 127, 183, 78	⊢
	: , : , : , : , : , :
70	instantiation	123, 79, 222, 124, 80, 81, 125, 101, 127, 183, 129, 82	⊢
	: , : , : , : , : , :
71	instantiation	177, 83, 84	⊢
	: , : , :
72	theorem		⊢
	proveit.numbers.rounding.ceil_of_real_above_int
73	instantiation	85, 231, 105, 86	⊢
	: , :
74	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
75	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
76	instantiation	87, 88, 200, 89	⊢
	: , :
77	instantiation	90, 149, 91, 111, 92, 93*	⊢
	: , : , :
78	instantiation	249, 211, 94	⊢
	: , : , :
79	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
80	instantiation	95	⊢
	: , : , :
81	instantiation	202	⊢
	: , :
82	instantiation	96, 183	⊢
	:
83	instantiation	97, 222, 251, 124, 98, 125, 101, 127, 183, 129, 99	⊢
	: , : , : , : , : , : , : , :
84	instantiation	100, 129, 101, 131	⊢
	: , : , :
85	theorem		⊢
	proveit.numbers.logarithms.log_base_large_a_greater_one
86	instantiation	102, 223, 103	⊢
	: , :
87	theorem		⊢
	proveit.numbers.rounding.ceil_increasing_less_eq
88	instantiation	206, 231, 105, 208	⊢
	: , :
89	instantiation	104, 231, 105, 207, 106, 223	⊢
	: , : , :
90	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
91	instantiation	109, 172, 150	⊢
	: , :
92	axiom		⊢
	proveit.physics.quantum.QPE._t_req
93	instantiation	177, 107, 108	⊢
	: , : , :
94	instantiation	109, 172, 110	⊢
	: , :
95	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
96	theorem		⊢
	proveit.numbers.negation.complex_closure
97	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_general
98	instantiation	202	⊢
	: , :
99	instantiation	151	⊢
	:
100	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
101	instantiation	249, 211, 111	⊢
	: , : , :
102	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
103	instantiation	112, 196, 113, 114, 115, 116*, 117*	⊢
	: , : , :
104	theorem		⊢
	proveit.numbers.logarithms.log_increasing_less_eq
105	instantiation	249, 233, 118	⊢
	: , : , :
106	instantiation	119, 196, 190, 120, 121, 122*	⊢
	: , : , :
107	instantiation	123, 124, 222, 251, 125, 126, 129, 130, 127	⊢
	: , : , : , : , : , :
108	instantiation	128, 129, 130, 131	⊢
	: , : , :
109	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
110	instantiation	171, 196	⊢
	:
111	instantiation	186, 187, 132	⊢
	: , : , :
112	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
113	instantiation	133, 190, 242	⊢
	: , :
114	instantiation	249, 245, 134	⊢
	: , : , :
115	instantiation	135, 190, 242, 243, 136, 137	⊢
	: , : , :
116	instantiation	177, 138, 139	⊢
	: , : , :
117	instantiation	177, 140, 141	⊢
	: , : , :
118	instantiation	217, 142, 234	⊢
	: , :
119	theorem		⊢
	proveit.numbers.addition.weak_bound_via_right_term_bound
120	instantiation	249, 143, 214	⊢
	: , : , :
121	instantiation	144, 145, 229, 231, 146	⊢
	: , : , :
122	instantiation	158, 212, 248, 159*, 147*, 148*	⊢
	: , : , : , :
123	theorem		⊢
	proveit.numbers.addition.disassociation
124	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
125	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
126	instantiation	202	⊢
	: , :
127	instantiation	249, 211, 149	⊢
	: , : , :
128	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
129	instantiation	249, 211, 172	⊢
	: , : , :
130	instantiation	249, 211, 150	⊢
	: , : , :
131	instantiation	151	⊢
	:
132	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
133	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
134	instantiation	152, 201, 246	⊢
	: , :
135	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_right_factor_bound
136	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
137	instantiation	153, 210	⊢
	:
138	instantiation	156, 154	⊢
	: , : , :
139	instantiation	155, 183	⊢
	:
140	instantiation	156, 157	⊢
	: , : , :
141	instantiation	158, 248, 212, 159*, 160*, 167*	⊢
	: , : , : , :
142	instantiation	249, 238, 161	⊢
	: , : , :
143	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
144	theorem		⊢
	proveit.numbers.division.weak_div_from_denom_bound__all_pos
145	instantiation	249, 162, 163	⊢
	: , : , :
146	instantiation	164, 196, 236, 243, 165, 166, 167*	⊢
	: , : , :
147	instantiation	177, 168, 169	⊢
	: , : , :
148	instantiation	170, 183	⊢
	:
149	instantiation	171, 172	⊢
	:
150	instantiation	249, 245, 173	⊢
	: , : , :
151	axiom		⊢
	proveit.logic.equality.equals_reflexivity
152	theorem		⊢
	proveit.numbers.multiplication.mult_rational_closure_bin
153	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
154	instantiation	174, 175	⊢
	:
155	theorem		⊢
	proveit.numbers.addition.elim_zero_left
156	axiom		⊢
	proveit.logic.equality.substitution
157	instantiation	203, 175	⊢
	:
158	theorem		⊢
	proveit.numbers.addition.rational_pair_addition
159	instantiation	176, 183	⊢
	:
160	instantiation	177, 178, 179	⊢
	: , : , :
161	theorem		⊢
	proveit.numbers.numerals.decimals.posnat5
162	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
163	instantiation	249, 180, 251	⊢
	: , : , :
164	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
165	instantiation	181, 242, 243, 244	⊢
	: , : , :
166	instantiation	182, 222	⊢
	:
167	instantiation	203, 183	⊢
	:
168	instantiation	191, 222, 184, 185, 195, 194	⊢
	: , : , : , :
169	theorem		⊢
	proveit.numbers.numerals.decimals.add_4_1
170	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
171	theorem		⊢
	proveit.numbers.negation.real_closure
172	instantiation	186, 187, 188	⊢
	: , : , :
173	instantiation	249, 247, 189	⊢
	: , : , :
174	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
175	instantiation	249, 211, 190	⊢
	: , : , :
176	theorem		⊢
	proveit.numbers.division.frac_one_denom
177	axiom		⊢
	proveit.logic.equality.equals_transitivity
178	instantiation	191, 222, 192, 193, 194, 195	⊢
	: , : , : , :
179	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_4
180	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
181	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_upper_bound
182	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_lower_bound
183	instantiation	249, 211, 196	⊢
	: , : , :
184	instantiation	202	⊢
	: , :
185	instantiation	202	⊢
	: , :
186	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
187	instantiation	197, 198	⊢
	: , :
188	axiom		⊢
	proveit.physics.quantum.QPE._n_in_natural_pos
189	instantiation	199, 200	⊢
	:
190	instantiation	249, 245, 201	⊢
	: , : , :
191	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
192	instantiation	202	⊢
	: , :
193	instantiation	202	⊢
	: , :
194	instantiation	203, 204	⊢
	:
195	theorem		⊢
	proveit.numbers.numerals.decimals.mult_2_2
196	instantiation	249, 245, 205	⊢
	: , : , :
197	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
198	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
199	axiom		⊢
	proveit.numbers.rounding.ceil_is_an_int
200	instantiation	206, 231, 207, 208	⊢
	: , :
201	instantiation	249, 209, 210	⊢
	: , : , :
202	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
203	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
204	instantiation	249, 211, 243	⊢
	: , : , :
205	instantiation	249, 247, 212	⊢
	: , : , :
206	theorem		⊢
	proveit.numbers.logarithms.log_real_pos_real_closure
207	instantiation	213, 231, 214	⊢
	: , :
208	instantiation	215, 216	⊢
	: , :
209	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
210	instantiation	217, 224, 234	⊢
	: , :
211	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
212	instantiation	249, 250, 222	⊢
	: , : , :
213	theorem		⊢
	proveit.numbers.addition.add_real_pos_closure_bin
214	instantiation	218, 219, 229, 220	⊢
	: , :
215	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
216	instantiation	221, 251, 222, 223	⊢
	: , :
217	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
218	theorem		⊢
	proveit.numbers.division.div_real_pos_closure
219	instantiation	249, 233, 224	⊢
	: , : , :
220	instantiation	225, 226	⊢
	:
221	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq_nat
222	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
223	theorem		⊢
	proveit.numbers.numerals.decimals.less_1_2
224	instantiation	249, 238, 227	⊢
	: , : , :
225	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
226	instantiation	249, 228, 229	⊢
	: , : , :
227	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
228	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
229	instantiation	230, 231, 232	⊢
	: , :
230	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
231	instantiation	249, 233, 234	⊢
	: , : , :
232	instantiation	235, 236, 237	⊢
	:
233	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
234	instantiation	249, 238, 239	⊢
	: , : , :
235	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pos_real_is_real_pos
236	instantiation	240, 242, 243, 244	⊢
	: , : , :
237	instantiation	241, 242, 243, 244	⊢
	: , : , :
238	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
239	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
240	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
241	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
242	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
243	instantiation	249, 245, 246	⊢
	: , : , :
244	axiom		⊢
	proveit.physics.quantum.QPE._eps_in_interval
245	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
246	instantiation	249, 247, 248	⊢
	: , : , :
247	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
248	instantiation	249, 250, 251	⊢
	: , : , :
249	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
250	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
251	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements