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