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