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*	⊢
	: , : , :
1	reference	88	⊢
2	reference	113	⊢
3	reference	17	⊢
4	instantiation	112, 26, 244	⊢
	: , :
5	instantiation	55, 17, 26, 244, 8, 9	⊢
	: , : , :
6	instantiation	10, 187, 11, 12	⊢
	: , : , :
7	instantiation	174, 13, 14	⊢
	: , : , :
8	instantiation	15, 198, 56, 224, 57, 16*	⊢
	: , : , :
9	instantiation	186, 252	⊢
	:
10	theorem		⊢
	proveit.numbers.addition.subtraction.subtract_from_add
11	instantiation	250, 209, 17	⊢
	: , : , :
12	theorem		⊢
	proveit.numbers.numerals.decimals.add_2_2
13	instantiation	130, 131, 223, 252, 132, 18, 21, 201, 79	⊢
	: , : , : , : , : , :
14	instantiation	19, 252, 223, 131, 20, 132, 21, 201, 79, 22*	⊢
	: , : , : , : , : , :
15	theorem		⊢
	proveit.numbers.exponentiation.exp_monotonicity_large_base_less_eq
16	instantiation	65, 23, 24, 191	⊢
	: , : , : , :
17	instantiation	250, 246, 25	⊢
	: , : , :
18	instantiation	199	⊢
	: , :
19	theorem		⊢
	proveit.numbers.addition.association
20	instantiation	199	⊢
	: , :
21	instantiation	250, 209, 26	⊢
	: , : , :
22	instantiation	27, 201, 187, 71	⊢
	: , : , :
23	instantiation	28, 240, 187	⊢
	: , :
24	instantiation	29, 223, 30, 31, 32	⊢
	: , : , : , :
25	instantiation	250, 248, 33	⊢
	: , : , :
26	instantiation	34, 198, 35	⊢
	: , :
27	theorem		⊢
	proveit.numbers.addition.subtraction.subtract_from_add_reversed
28	theorem		⊢
	proveit.numbers.exponentiation.exp_nat_pos_expansion
29	theorem		⊢
	proveit.core_expr_types.operations.operands_substitution_via_tuple
30	instantiation	36, 223	⊢
	: , :
31	instantiation	199	⊢
	: , :
32	instantiation	37	⊢
	:
33	instantiation	250, 251, 38	⊢
	: , : , :
34	theorem		⊢
	proveit.numbers.exponentiation.exp_real_closure_nat_power
35	instantiation	39, 40, 41	⊢
	:
36	theorem		⊢
	proveit.core_expr_types.tuples.range_from1_len_typical_eq
37	theorem		⊢
	proveit.numbers.numerals.decimals.reduce_2_repeats
38	theorem		⊢
	proveit.numbers.numerals.decimals.nat4
39	theorem		⊢
	proveit.numbers.number_sets.integers.nonneg_int_is_natural
40	instantiation	42, 43, 44	⊢
	: , :
41	instantiation	45, 46	⊢
	: , :
42	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
43	instantiation	250, 47, 139	⊢
	: , : , :
44	instantiation	250, 48, 49	⊢
	: , : , :
45	theorem		⊢
	proveit.numbers.addition.subtraction.nonneg_difference
46	instantiation	88, 92, 198, 50, 51, 52*, 53*	⊢
	: , : , :
47	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
48	theorem		⊢
	proveit.numbers.number_sets.integers.neg_int_within_int
49	instantiation	54, 195	⊢
	:
50	instantiation	112, 56, 198	⊢
	: , :
51	instantiation	55, 198, 56, 57, 171	⊢
	: , : , :
52	instantiation	174, 58, 59	⊢
	: , : , :
53	instantiation	174, 60, 61	⊢
	: , : , :
54	theorem		⊢
	proveit.numbers.negation.int_neg_closure
55	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right
56	instantiation	112, 114, 158	⊢
	: , :
57	instantiation	72, 62, 63	⊢
	: , : , :
58	instantiation	130, 252, 223, 131, 78, 132, 187, 136, 79	⊢
	: , : , : , : , : , :
59	instantiation	135, 187, 136, 97	⊢
	: , : , :
60	instantiation	144, 64	⊢
	: , : , :
61	instantiation	65, 66, 67, 68	⊢
	: , : , : , :
62	instantiation	69, 249, 86, 70, 71*	⊢
	: , :
63	instantiation	72, 73, 74	⊢
	: , : , :
64	instantiation	130, 131, 223, 252, 132, 96, 99, 134, 187	⊢
	: , : , : , : , : , :
65	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
66	instantiation	130, 131, 76, 252, 132, 77, 99, 134, 187, 75	⊢
	: , : , : , : , : , :
67	instantiation	130, 76, 223, 131, 77, 78, 132, 99, 134, 187, 136, 79	⊢
	: , : , : , : , : , :
68	instantiation	174, 80, 81	⊢
	: , : , :
69	theorem		⊢
	proveit.numbers.rounding.ceil_of_real_above_int
70	instantiation	82, 232, 108, 83, 224, 84*	⊢
	: , : , :
71	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
72	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
73	instantiation	85, 86, 205, 87	⊢
	: , :
74	instantiation	88, 158, 89, 114, 90, 91*	⊢
	: , : , :
75	instantiation	250, 209, 92	⊢
	: , : , :
76	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
77	instantiation	93	⊢
	: , : , :
78	instantiation	199	⊢
	: , :
79	instantiation	94, 187	⊢
	:
80	instantiation	95, 223, 252, 131, 96, 132, 99, 134, 187, 136, 97	⊢
	: , : , : , : , : , : , : , :
81	instantiation	98, 136, 99, 138	⊢
	: , : , :
82	theorem		⊢
	proveit.numbers.logarithms.log_increasing_less
83	instantiation	100, 198, 101, 102, 103, 104*, 105*	⊢
	: , : , :
84	instantiation	106, 232	⊢
	:
85	theorem		⊢
	proveit.numbers.rounding.ceil_increasing_less_eq
86	instantiation	210, 232, 108, 212	⊢
	: , :
87	instantiation	107, 232, 108, 211, 109, 224	⊢
	: , : , :
88	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
89	instantiation	112, 179, 159	⊢
	: , :
90	axiom		⊢
	proveit.physics.quantum.QPE._t_req
91	instantiation	174, 110, 111	⊢
	: , : , :
92	instantiation	112, 179, 113	⊢
	: , :
93	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
94	theorem		⊢
	proveit.numbers.negation.complex_closure
95	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_general
96	instantiation	199	⊢
	: , :
97	instantiation	160	⊢
	:
98	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
99	instantiation	250, 209, 114	⊢
	: , : , :
100	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
101	instantiation	115, 181, 243	⊢
	: , :
102	instantiation	250, 246, 116	⊢
	: , : , :
103	instantiation	117, 181, 243, 244, 118, 119	⊢
	: , : , :
104	instantiation	174, 120, 121	⊢
	: , : , :
105	instantiation	174, 122, 123	⊢
	: , : , :
106	theorem		⊢
	proveit.numbers.logarithms.log_eq_1
107	theorem		⊢
	proveit.numbers.logarithms.log_increasing_less_eq
108	instantiation	219, 124, 232, 149	⊢
	: , :
109	instantiation	125, 198, 126, 127, 128, 129*	⊢
	: , : , :
110	instantiation	130, 131, 223, 252, 132, 133, 136, 137, 134	⊢
	: , : , : , : , : , :
111	instantiation	135, 136, 137, 138	⊢
	: , : , :
112	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
113	instantiation	178, 198	⊢
	:
114	instantiation	193, 194, 139	⊢
	: , : , :
115	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
116	instantiation	140, 197, 247	⊢
	: , :
117	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_right_factor_bound
118	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
119	instantiation	141, 207	⊢
	:
120	instantiation	144, 142	⊢
	: , : , :
121	instantiation	143, 187	⊢
	:
122	instantiation	144, 145	⊢
	: , : , :
123	instantiation	154, 249, 214, 155*, 146*, 172*	⊢
	: , : , : , :
124	instantiation	250, 234, 147	⊢
	: , : , :
125	theorem		⊢
	proveit.numbers.addition.weak_bound_via_right_term_bound
126	instantiation	148, 244, 198, 149	⊢
	: , :
127	instantiation	250, 150, 216	⊢
	: , : , :
128	instantiation	151, 152, 230, 232, 153	⊢
	: , : , :
129	instantiation	154, 214, 249, 155*, 156*, 157*	⊢
	: , : , : , :
130	theorem		⊢
	proveit.numbers.addition.disassociation
131	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
132	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
133	instantiation	199	⊢
	: , :
134	instantiation	250, 209, 158	⊢
	: , : , :
135	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
136	instantiation	250, 209, 179	⊢
	: , : , :
137	instantiation	250, 209, 159	⊢
	: , : , :
138	instantiation	160	⊢
	:
139	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
140	theorem		⊢
	proveit.numbers.multiplication.mult_rational_closure_bin
141	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
142	instantiation	161, 162	⊢
	:
143	theorem		⊢
	proveit.numbers.addition.elim_zero_left
144	axiom		⊢
	proveit.logic.equality.substitution
145	instantiation	200, 162	⊢
	:
146	instantiation	174, 163, 164	⊢
	: , : , :
147	instantiation	250, 239, 165	⊢
	: , : , :
148	theorem		⊢
	proveit.numbers.division.div_real_closure
149	instantiation	166, 240	⊢
	:
150	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
151	theorem		⊢
	proveit.numbers.division.weak_div_from_denom_bound__all_pos
152	instantiation	250, 167, 168	⊢
	: , : , :
153	instantiation	169, 198, 237, 244, 170, 171, 172*	⊢
	: , : , :
154	theorem		⊢
	proveit.numbers.addition.rational_pair_addition
155	instantiation	173, 187	⊢
	:
156	instantiation	174, 175, 176	⊢
	: , : , :
157	instantiation	177, 187	⊢
	:
158	instantiation	178, 179	⊢
	:
159	instantiation	250, 246, 180	⊢
	: , : , :
160	axiom		⊢
	proveit.logic.equality.equals_reflexivity
161	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
162	instantiation	250, 209, 181	⊢
	: , : , :
163	instantiation	188, 223, 182, 183, 192, 191	⊢
	: , : , : , :
164	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_4
165	theorem		⊢
	proveit.numbers.numerals.decimals.posnat5
166	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
167	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
168	instantiation	250, 184, 252	⊢
	: , : , :
169	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
170	instantiation	185, 243, 244, 245	⊢
	: , : , :
171	instantiation	186, 223	⊢
	:
172	instantiation	200, 187	⊢
	:
173	theorem		⊢
	proveit.numbers.division.frac_one_denom
174	axiom		⊢
	proveit.logic.equality.equals_transitivity
175	instantiation	188, 223, 189, 190, 191, 192	⊢
	: , : , : , :
176	theorem		⊢
	proveit.numbers.numerals.decimals.add_4_1
177	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
178	theorem		⊢
	proveit.numbers.negation.real_closure
179	instantiation	193, 194, 195	⊢
	: , : , :
180	instantiation	250, 248, 196	⊢
	: , : , :
181	instantiation	250, 246, 197	⊢
	: , : , :
182	instantiation	199	⊢
	: , :
183	instantiation	199	⊢
	: , :
184	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
185	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_upper_bound
186	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_lower_bound
187	instantiation	250, 209, 198	⊢
	: , : , :
188	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
189	instantiation	199	⊢
	: , :
190	instantiation	199	⊢
	: , :
191	theorem		⊢
	proveit.numbers.numerals.decimals.mult_2_2
192	instantiation	200, 201	⊢
	:
193	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
194	instantiation	202, 203	⊢
	: , :
195	axiom		⊢
	proveit.physics.quantum.QPE._n_in_natural_pos
196	instantiation	204, 205	⊢
	:
197	instantiation	250, 206, 207	⊢
	: , : , :
198	instantiation	250, 246, 208	⊢
	: , : , :
199	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
200	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
201	instantiation	250, 209, 244	⊢
	: , : , :
202	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
203	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
204	axiom		⊢
	proveit.numbers.rounding.ceil_is_an_int
205	instantiation	210, 232, 211, 212	⊢
	: , :
206	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
207	instantiation	213, 225, 235	⊢
	: , :
208	instantiation	250, 248, 214	⊢
	: , : , :
209	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
210	theorem		⊢
	proveit.numbers.logarithms.log_real_pos_real_closure
211	instantiation	215, 232, 216	⊢
	: , :
212	instantiation	217, 218	⊢
	: , :
213	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
214	instantiation	250, 251, 223	⊢
	: , : , :
215	theorem		⊢
	proveit.numbers.addition.add_real_pos_closure_bin
216	instantiation	219, 220, 230, 221	⊢
	: , :
217	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
218	instantiation	222, 252, 223, 224	⊢
	: , :
219	theorem		⊢
	proveit.numbers.division.div_real_pos_closure
220	instantiation	250, 234, 225	⊢
	: , : , :
221	instantiation	226, 227	⊢
	:
222	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq_nat
223	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
224	theorem		⊢
	proveit.numbers.numerals.decimals.less_1_2
225	instantiation	250, 239, 228	⊢
	: , : , :
226	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
227	instantiation	250, 229, 230	⊢
	: , : , :
228	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
229	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
230	instantiation	231, 232, 233	⊢
	: , :
231	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
232	instantiation	250, 234, 235	⊢
	: , : , :
233	instantiation	236, 237, 238	⊢
	:
234	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
235	instantiation	250, 239, 240	⊢
	: , : , :
236	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pos_real_is_real_pos
237	instantiation	241, 243, 244, 245	⊢
	: , : , :
238	instantiation	242, 243, 244, 245	⊢
	: , : , :
239	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
240	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
241	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
242	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
243	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
244	instantiation	250, 246, 247	⊢
	: , : , :
245	axiom		⊢
	proveit.physics.quantum.QPE._eps_in_interval
246	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
247	instantiation	250, 248, 249	⊢
	: , : , :
248	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
249	instantiation	250, 251, 252	⊢
	: , : , :
250	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
251	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
252	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements