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