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, 10*	⊢
	: , : , : , : , : , :
1	theorem		⊢
	proveit.numbers.addition.association
2	reference	228	⊢
3	reference	199	⊢
4	reference	107	⊢
5	instantiation	175	⊢
	: , :
6	reference	108	⊢
7	instantiation	226, 185, 11	⊢
	: , : , :
8	reference	177	⊢
9	reference	55	⊢
10	instantiation	12, 177, 163, 47	⊢
	: , : , :
11	instantiation	13, 174, 14	⊢
	: , :
12	theorem		⊢
	proveit.numbers.addition.subtraction.subtract_from_add_reversed
13	theorem		⊢
	proveit.numbers.exponentiation.exp_real_closure_nat_power
14	instantiation	15, 16, 17	⊢
	:
15	theorem		⊢
	proveit.numbers.number_sets.integers.nonneg_int_is_natural
16	instantiation	18, 19, 20	⊢
	: , :
17	instantiation	21, 22	⊢
	: , :
18	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
19	instantiation	226, 23, 115	⊢
	: , : , :
20	instantiation	226, 24, 25	⊢
	: , : , :
21	theorem		⊢
	proveit.numbers.addition.subtraction.nonneg_difference
22	instantiation	64, 68, 174, 26, 27, 28*, 29*	⊢
	: , : , :
23	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
24	theorem		⊢
	proveit.numbers.number_sets.integers.neg_int_within_int
25	instantiation	30, 171	⊢
	:
26	instantiation	88, 32, 174	⊢
	: , :
27	instantiation	31, 174, 32, 33, 147	⊢
	: , : , :
28	instantiation	150, 34, 35	⊢
	: , : , :
29	instantiation	150, 36, 37	⊢
	: , : , :
30	theorem		⊢
	proveit.numbers.negation.int_neg_closure
31	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right
32	instantiation	88, 90, 134	⊢
	: , :
33	instantiation	48, 38, 39	⊢
	: , : , :
34	instantiation	106, 228, 199, 107, 54, 108, 163, 112, 55	⊢
	: , : , : , : , : , :
35	instantiation	111, 163, 112, 73	⊢
	: , : , :
36	instantiation	120, 40	⊢
	: , : , :
37	instantiation	41, 42, 43, 44	⊢
	: , : , : , :
38	instantiation	45, 225, 62, 46, 47*	⊢
	: , :
39	instantiation	48, 49, 50	⊢
	: , : , :
40	instantiation	106, 107, 199, 228, 108, 72, 75, 110, 163	⊢
	: , : , : , : , : , :
41	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
42	instantiation	106, 107, 52, 228, 108, 53, 75, 110, 163, 51	⊢
	: , : , : , : , : , :
43	instantiation	106, 52, 199, 107, 53, 54, 108, 75, 110, 163, 112, 55	⊢
	: , : , : , : , : , :
44	instantiation	150, 56, 57	⊢
	: , : , :
45	theorem		⊢
	proveit.numbers.rounding.ceil_of_real_above_int
46	instantiation	58, 208, 84, 59, 200, 60*	⊢
	: , : , :
47	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
48	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
49	instantiation	61, 62, 181, 63	⊢
	: , :
50	instantiation	64, 134, 65, 90, 66, 67*	⊢
	: , : , :
51	instantiation	226, 185, 68	⊢
	: , : , :
52	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
53	instantiation	69	⊢
	: , : , :
54	instantiation	175	⊢
	: , :
55	instantiation	70, 163	⊢
	:
56	instantiation	71, 199, 228, 107, 72, 108, 75, 110, 163, 112, 73	⊢
	: , : , : , : , : , : , : , :
57	instantiation	74, 112, 75, 114	⊢
	: , : , :
58	theorem		⊢
	proveit.numbers.logarithms.log_increasing_less
59	instantiation	76, 174, 77, 78, 79, 80*, 81*	⊢
	: , : , :
60	instantiation	82, 208	⊢
	:
61	theorem		⊢
	proveit.numbers.rounding.ceil_increasing_less_eq
62	instantiation	186, 208, 84, 188	⊢
	: , :
63	instantiation	83, 208, 84, 187, 85, 200	⊢
	: , : , :
64	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
65	instantiation	88, 155, 135	⊢
	: , :
66	axiom		⊢
	proveit.physics.quantum.QPE._t_req
67	instantiation	150, 86, 87	⊢
	: , : , :
68	instantiation	88, 155, 89	⊢
	: , :
69	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
70	theorem		⊢
	proveit.numbers.negation.complex_closure
71	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_general
72	instantiation	175	⊢
	: , :
73	instantiation	136	⊢
	:
74	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
75	instantiation	226, 185, 90	⊢
	: , : , :
76	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
77	instantiation	91, 157, 219	⊢
	: , :
78	instantiation	226, 222, 92	⊢
	: , : , :
79	instantiation	93, 157, 219, 220, 94, 95	⊢
	: , : , :
80	instantiation	150, 96, 97	⊢
	: , : , :
81	instantiation	150, 98, 99	⊢
	: , : , :
82	theorem		⊢
	proveit.numbers.logarithms.log_eq_1
83	theorem		⊢
	proveit.numbers.logarithms.log_increasing_less_eq
84	instantiation	195, 100, 208, 125	⊢
	: , :
85	instantiation	101, 174, 102, 103, 104, 105*	⊢
	: , : , :
86	instantiation	106, 107, 199, 228, 108, 109, 112, 113, 110	⊢
	: , : , : , : , : , :
87	instantiation	111, 112, 113, 114	⊢
	: , : , :
88	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
89	instantiation	154, 174	⊢
	:
90	instantiation	169, 170, 115	⊢
	: , : , :
91	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
92	instantiation	116, 173, 223	⊢
	: , :
93	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_right_factor_bound
94	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
95	instantiation	117, 183	⊢
	:
96	instantiation	120, 118	⊢
	: , : , :
97	instantiation	119, 163	⊢
	:
98	instantiation	120, 121	⊢
	: , : , :
99	instantiation	130, 225, 190, 131*, 122*, 148*	⊢
	: , : , : , :
100	instantiation	226, 210, 123	⊢
	: , : , :
101	theorem		⊢
	proveit.numbers.addition.weak_bound_via_right_term_bound
102	instantiation	124, 220, 174, 125	⊢
	: , :
103	instantiation	226, 126, 192	⊢
	: , : , :
104	instantiation	127, 128, 206, 208, 129	⊢
	: , : , :
105	instantiation	130, 190, 225, 131*, 132*, 133*	⊢
	: , : , : , :
106	theorem		⊢
	proveit.numbers.addition.disassociation
107	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
108	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
109	instantiation	175	⊢
	: , :
110	instantiation	226, 185, 134	⊢
	: , : , :
111	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
112	instantiation	226, 185, 155	⊢
	: , : , :
113	instantiation	226, 185, 135	⊢
	: , : , :
114	instantiation	136	⊢
	:
115	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
116	theorem		⊢
	proveit.numbers.multiplication.mult_rational_closure_bin
117	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
118	instantiation	137, 138	⊢
	:
119	theorem		⊢
	proveit.numbers.addition.elim_zero_left
120	axiom		⊢
	proveit.logic.equality.substitution
121	instantiation	176, 138	⊢
	:
122	instantiation	150, 139, 140	⊢
	: , : , :
123	instantiation	226, 215, 141	⊢
	: , : , :
124	theorem		⊢
	proveit.numbers.division.div_real_closure
125	instantiation	142, 216	⊢
	:
126	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
127	theorem		⊢
	proveit.numbers.division.weak_div_from_denom_bound__all_pos
128	instantiation	226, 143, 144	⊢
	: , : , :
129	instantiation	145, 174, 213, 220, 146, 147, 148*	⊢
	: , : , :
130	theorem		⊢
	proveit.numbers.addition.rational_pair_addition
131	instantiation	149, 163	⊢
	:
132	instantiation	150, 151, 152	⊢
	: , : , :
133	instantiation	153, 163	⊢
	:
134	instantiation	154, 155	⊢
	:
135	instantiation	226, 222, 156	⊢
	: , : , :
136	axiom		⊢
	proveit.logic.equality.equals_reflexivity
137	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
138	instantiation	226, 185, 157	⊢
	: , : , :
139	instantiation	164, 199, 158, 159, 168, 167	⊢
	: , : , : , :
140	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_4
141	theorem		⊢
	proveit.numbers.numerals.decimals.posnat5
142	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
143	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
144	instantiation	226, 160, 228	⊢
	: , : , :
145	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
146	instantiation	161, 219, 220, 221	⊢
	: , : , :
147	instantiation	162, 199	⊢
	:
148	instantiation	176, 163	⊢
	:
149	theorem		⊢
	proveit.numbers.division.frac_one_denom
150	axiom		⊢
	proveit.logic.equality.equals_transitivity
151	instantiation	164, 199, 165, 166, 167, 168	⊢
	: , : , : , :
152	theorem		⊢
	proveit.numbers.numerals.decimals.add_4_1
153	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
154	theorem		⊢
	proveit.numbers.negation.real_closure
155	instantiation	169, 170, 171	⊢
	: , : , :
156	instantiation	226, 224, 172	⊢
	: , : , :
157	instantiation	226, 222, 173	⊢
	: , : , :
158	instantiation	175	⊢
	: , :
159	instantiation	175	⊢
	: , :
160	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
161	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_upper_bound
162	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_lower_bound
163	instantiation	226, 185, 174	⊢
	: , : , :
164	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
165	instantiation	175	⊢
	: , :
166	instantiation	175	⊢
	: , :
167	theorem		⊢
	proveit.numbers.numerals.decimals.mult_2_2
168	instantiation	176, 177	⊢
	:
169	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
170	instantiation	178, 179	⊢
	: , :
171	axiom		⊢
	proveit.physics.quantum.QPE._n_in_natural_pos
172	instantiation	180, 181	⊢
	:
173	instantiation	226, 182, 183	⊢
	: , : , :
174	instantiation	226, 222, 184	⊢
	: , : , :
175	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
176	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
177	instantiation	226, 185, 220	⊢
	: , : , :
178	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
179	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
180	axiom		⊢
	proveit.numbers.rounding.ceil_is_an_int
181	instantiation	186, 208, 187, 188	⊢
	: , :
182	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
183	instantiation	189, 201, 211	⊢
	: , :
184	instantiation	226, 224, 190	⊢
	: , : , :
185	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
186	theorem		⊢
	proveit.numbers.logarithms.log_real_pos_real_closure
187	instantiation	191, 208, 192	⊢
	: , :
188	instantiation	193, 194	⊢
	: , :
189	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
190	instantiation	226, 227, 199	⊢
	: , : , :
191	theorem		⊢
	proveit.numbers.addition.add_real_pos_closure_bin
192	instantiation	195, 196, 206, 197	⊢
	: , :
193	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
194	instantiation	198, 228, 199, 200	⊢
	: , :
195	theorem		⊢
	proveit.numbers.division.div_real_pos_closure
196	instantiation	226, 210, 201	⊢
	: , : , :
197	instantiation	202, 203	⊢
	:
198	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq_nat
199	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
200	theorem		⊢
	proveit.numbers.numerals.decimals.less_1_2
201	instantiation	226, 215, 204	⊢
	: , : , :
202	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
203	instantiation	226, 205, 206	⊢
	: , : , :
204	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
205	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
206	instantiation	207, 208, 209	⊢
	: , :
207	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
208	instantiation	226, 210, 211	⊢
	: , : , :
209	instantiation	212, 213, 214	⊢
	:
210	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
211	instantiation	226, 215, 216	⊢
	: , : , :
212	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pos_real_is_real_pos
213	instantiation	217, 219, 220, 221	⊢
	: , : , :
214	instantiation	218, 219, 220, 221	⊢
	: , : , :
215	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
216	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
217	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
218	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
219	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
220	instantiation	226, 222, 223	⊢
	: , : , :
221	axiom		⊢
	proveit.physics.quantum.QPE._eps_in_interval
222	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
223	instantiation	226, 224, 225	⊢
	: , : , :
224	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
225	instantiation	226, 227, 228	⊢
	: , : , :
226	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
227	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
228	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements