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