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	reference	262	⊢
2	instantiation	4, 268, 5, 6, 7	⊢
	: , : , :
3	instantiation	8, 9	⊢
	: , :
4	theorem		⊢
	proveit.logic.sets.unification.union_inclusion
5	instantiation	216	⊢
	: , :
6	instantiation	12, 53, 10, 280, 11	⊢
	: , : , : , :
7	instantiation	12, 53, 14, 280, 13	⊢
	: , : , : , :
8	theorem		⊢
	proveit.logic.booleans.conjunction.left_from_and
9	assumption		⊢
10	instantiation	286, 14	⊢
	:
11	instantiation	18, 212, 15, 16, 27, 17	⊢
	: , :
12	theorem		⊢
	proveit.numbers.number_sets.integers.interval_subset_eq
13	instantiation	18, 212, 19, 20, 21, 22	⊢
	: , :
14	instantiation	279, 255, 276	⊢
	: , :
15	instantiation	223	⊢
	: , : , :
16	instantiation	32, 23, 24	⊢
	: , :
17	instantiation	153, 25	⊢
	: , :
18	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_all
19	instantiation	223	⊢
	: , : , :
20	instantiation	26, 27, 28	⊢
	: , : , :
21	instantiation	88, 258, 237, 29, 30, 31*	⊢
	: , : , :
22	instantiation	32, 33, 34	⊢
	: , :
23	instantiation	288, 277, 35	⊢
	: , : , :
24	instantiation	203	⊢
	:
25	instantiation	36, 37, 38	⊢
	: , : , :
26	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
27	instantiation	88, 39, 237, 48, 49, 40*, 41*	⊢
	: , : , :
28	instantiation	88, 120, 42, 43, 44, 45*, 46*	⊢
	: , : , :
29	instantiation	141, 48, 258	⊢
	: , :
30	instantiation	47, 237, 48, 258, 49, 50	⊢
	: , : , :
31	instantiation	207, 51, 190, 52	⊢
	: , : , : , :
32	theorem		⊢
	proveit.numbers.ordering.relax_equal_to_less_eq
33	instantiation	262, 263, 285	⊢
	: , : , :
34	instantiation	203	⊢
	:
35	instantiation	288, 283, 53	⊢
	: , : , :
36	axiom		⊢
	proveit.numbers.ordering.transitivity_less_less
37	instantiation	54, 55	⊢
	:
38	instantiation	112, 285	⊢
	:
39	instantiation	56, 212, 57, 58, 258, 142	⊢
	: , :
40	instantiation	207, 59, 60, 61	⊢
	: , : , : , :
41	instantiation	219, 62	⊢
	: , :
42	instantiation	159, 271, 89	⊢
	: , :
43	instantiation	141, 143, 271	⊢
	: , :
44	instantiation	63, 64, 65, 287, 66, 67	⊢
	: , : , :
45	instantiation	219, 68	⊢
	: , :
46	instantiation	207, 69, 70, 71	⊢
	: , : , : , :
47	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right
48	instantiation	288, 277, 72	⊢
	: , : , :
49	instantiation	73, 276, 275, 266	⊢
	: , : , :
50	instantiation	113, 282	⊢
	:
51	instantiation	185, 74, 75	⊢
	: , : , :
52	instantiation	219, 197	⊢
	: , :
53	instantiation	279, 106, 276	⊢
	: , :
54	theorem		⊢
	proveit.numbers.number_sets.integers.negative_if_in_neg_int
55	instantiation	76, 77	⊢
	:
56	theorem		⊢
	proveit.numbers.addition.add_real_closure
57	instantiation	223	⊢
	: , : , :
58	instantiation	288, 277, 78	⊢
	: , : , :
59	instantiation	185, 79, 80	⊢
	: , : , :
60	instantiation	203	⊢
	:
61	instantiation	219, 81	⊢
	: , :
62	instantiation	207, 93, 82, 83	⊢
	: , : , : , :
63	theorem		⊢
	proveit.numbers.ordering.less_add_right_weak_int
64	instantiation	288, 289, 84	⊢
	: , : , :
65	instantiation	288, 289, 85	⊢
	: , : , :
66	instantiation	86, 271, 89, 237, 154, 87	⊢
	: , : , :
67	instantiation	88, 230, 89, 138, 90, 91*, 92*	⊢
	: , : , :
68	instantiation	207, 93, 94, 95	⊢
	: , : , : , :
69	instantiation	199, 200, 290, 282, 202, 97, 123, 261, 96	⊢
	: , : , : , : , : , :
70	instantiation	199, 290, 200, 97, 178, 202, 123, 261, 215, 228	⊢
	: , : , : , : , : , :
71	instantiation	207, 98, 99, 100	⊢
	: , : , : , :
72	instantiation	288, 283, 275	⊢
	: , : , :
73	theorem		⊢
	proveit.numbers.number_sets.integers.interval_upper_bound
74	instantiation	221, 101	⊢
	: , : , :
75	instantiation	185, 102, 103	⊢
	: , : , :
76	theorem		⊢
	proveit.numbers.negation.int_neg_closure
77	instantiation	104, 105, 233	⊢
	: , :
78	instantiation	288, 283, 106	⊢
	: , : , :
79	instantiation	221, 165	⊢
	: , : , :
80	instantiation	185, 107, 108	⊢
	: , : , :
81	instantiation	221, 184	⊢
	: , : , :
82	instantiation	227, 215, 228	⊢
	: , :
83	instantiation	219, 109	⊢
	: , :
84	instantiation	110, 290, 200	⊢
	: , :
85	instantiation	110, 290, 111	⊢
	: , :
86	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_right_factor_bound
87	instantiation	112, 268	⊢
	:
88	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
89	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
90	instantiation	113, 212	⊢
	:
91	instantiation	191, 114, 115	⊢
	: , : , :
92	instantiation	196, 261, 250, 116, 117	⊢
	: , : , :
93	instantiation	118, 225, 250	⊢
	: , :
94	instantiation	203	⊢
	:
95	instantiation	219, 119	⊢
	: , :
96	instantiation	288, 270, 120	⊢
	: , : , :
97	instantiation	216	⊢
	: , :
98	instantiation	121, 282, 123, 261, 215, 228	⊢
	: , : , : , : , : , : , :
99	instantiation	168, 200, 290, 202, 122, 181, 123, 215, 261, 228, 124*	⊢
	: , : , : , : , : , :
100	instantiation	168, 282, 290, 200, 181, 202, 225, 261, 228, 182*	⊢
	: , : , : , : , : , :
101	instantiation	185, 125, 126	⊢
	: , : , :
102	instantiation	199, 200, 290, 282, 202, 127, 224, 228, 250	⊢
	: , : , : , : , : , :
103	instantiation	128, 250, 224, 129	⊢
	: , : , :
104	theorem		⊢
	proveit.numbers.addition.add_nat_pos_closure_bin
105	instantiation	130, 255, 154	⊢
	:
106	instantiation	286, 280	⊢
	:
107	instantiation	199, 282, 212, 200, 213, 202, 225, 214, 250, 215	⊢
	: , : , : , : , : , :
108	instantiation	188, 200, 290, 202, 131, 225, 214, 250, 132	⊢
	: , : , : , : , : , : , : , :
109	instantiation	185, 133, 134	⊢
	: , : , :
110	theorem		⊢
	proveit.numbers.multiplication.mult_nat_closure_bin
111	instantiation	135, 255, 136	⊢
	:
112	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
113	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_lower_bound
114	instantiation	137, 228	⊢
	:
115	instantiation	227, 228, 176	⊢
	: , :
116	instantiation	288, 270, 138	⊢
	: , : , :
117	theorem		⊢
	proveit.numbers.numerals.decimals.add_2_1
118	theorem		⊢
	proveit.numbers.negation.distribute_neg_through_binary_sum
119	instantiation	185, 139, 140	⊢
	: , : , :
120	instantiation	141, 142, 230	⊢
	: , :
121	theorem		⊢
	proveit.numbers.addition.leftward_commutation
122	instantiation	216	⊢
	: , :
123	instantiation	288, 270, 143	⊢
	: , : , :
124	instantiation	185, 144, 145, 146*	⊢
	: , : , :
125	instantiation	221, 164	⊢
	: , : , :
126	instantiation	185, 147, 148	⊢
	: , : , :
127	instantiation	216	⊢
	: , :
128	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
129	instantiation	203	⊢
	:
130	theorem		⊢
	proveit.numbers.number_sets.integers.pos_int_is_natural_pos
131	instantiation	216	⊢
	: , :
132	instantiation	203	⊢
	:
133	instantiation	185, 149, 150	⊢
	: , : , :
134	instantiation	185, 151, 152	⊢
	: , : , :
135	theorem		⊢
	proveit.numbers.number_sets.integers.nonneg_int_is_natural
136	instantiation	153, 154	⊢
	: , :
137	theorem		⊢
	proveit.numbers.addition.elim_zero_right
138	instantiation	288, 277, 155	⊢
	: , : , :
139	instantiation	221, 156	⊢
	: , : , :
140	instantiation	185, 157, 158	⊢
	: , : , :
141	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
142	instantiation	239, 237	⊢
	:
143	instantiation	159, 271, 237	⊢
	: , :
144	instantiation	221, 160	⊢
	: , : , :
145	instantiation	219, 161	⊢
	: , :
146	instantiation	185, 162, 163	⊢
	: , : , :
147	instantiation	199, 200, 290, 282, 202, 201, 224, 206, 250	⊢
	: , : , : , : , : , :
148	instantiation	168, 282, 290, 200, 169, 202, 224, 206, 250, 170*	⊢
	: , : , : , : , : , :
149	instantiation	221, 164	⊢
	: , : , :
150	instantiation	221, 165	⊢
	: , : , :
151	instantiation	185, 166, 167	⊢
	: , : , :
152	instantiation	168, 200, 290, 282, 202, 169, 206, 250, 215, 170*	⊢
	: , : , : , : , : , :
153	theorem		⊢
	proveit.numbers.ordering.relax_less
154	instantiation	171, 172, 173	⊢
	: , : , :
155	instantiation	288, 283, 174	⊢
	: , : , :
156	instantiation	175, 261	⊢
	:
157	instantiation	199, 282, 290, 200, 178, 202, 176, 215, 228	⊢
	: , : , : , : , : , :
158	instantiation	177, 200, 290, 202, 178, 215, 228	⊢
	: , : , : , :
159	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
160	instantiation	219, 179	⊢
	: , :
161	instantiation	180, 200, 290, 282, 202, 181, 261, 228, 225	⊢
	: , : , : , : , : , :
162	instantiation	221, 182	⊢
	: , : , :
163	instantiation	183, 225	⊢
	:
164	instantiation	221, 197	⊢
	: , : , :
165	instantiation	221, 184	⊢
	: , : , :
166	instantiation	185, 186, 187	⊢
	: , : , :
167	instantiation	188, 200, 282, 290, 202, 189, 224, 206, 250, 215, 190	⊢
	: , : , : , : , : , : , : , :
168	theorem		⊢
	proveit.numbers.addition.association
169	instantiation	216	⊢
	: , :
170	instantiation	191, 192, 193	⊢
	: , : , :
171	theorem		⊢
	proveit.numbers.ordering.transitivity_less_less_eq
172	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
173	instantiation	194, 276, 275, 266	⊢
	: , : , :
174	instantiation	288, 289, 212	⊢
	: , : , :
175	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
176	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.zero_is_complex
177	theorem		⊢
	proveit.numbers.addition.elim_zero_any
178	instantiation	216	⊢
	: , :
179	instantiation	195, 225	⊢
	:
180	theorem		⊢
	proveit.numbers.multiplication.distribute_through_sum
181	instantiation	216	⊢
	: , :
182	instantiation	196, 250, 261, 205	⊢
	: , : , :
183	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
184	instantiation	221, 197	⊢
	: , : , :
185	axiom		⊢
	proveit.logic.equality.equals_transitivity
186	instantiation	199, 200, 290, 282, 202, 201, 224, 206, 198	⊢
	: , : , : , : , : , :
187	instantiation	199, 290, 212, 200, 201, 213, 202, 224, 206, 214, 250, 215	⊢
	: , : , : , : , : , :
188	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_general
189	instantiation	216	⊢
	: , :
190	instantiation	203	⊢
	:
191	theorem		⊢
	proveit.logic.equality.sub_right_side_into
192	instantiation	204, 250, 261, 205	⊢
	: , : , :
193	instantiation	227, 250, 206	⊢
	: , :
194	theorem		⊢
	proveit.numbers.number_sets.integers.interval_lower_bound
195	theorem		⊢
	proveit.numbers.negation.mult_neg_one_left
196	theorem		⊢
	proveit.numbers.addition.subtraction.subtract_from_add
197	instantiation	207, 208, 209, 210	⊢
	: , : , : , :
198	instantiation	211, 212, 213, 214, 250, 215	⊢
	: , :
199	theorem		⊢
	proveit.numbers.addition.disassociation
200	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
201	instantiation	216	⊢
	: , :
202	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
203	axiom		⊢
	proveit.logic.equality.equals_reflexivity
204	theorem		⊢
	proveit.numbers.addition.subtraction.subtract_from_add_reversed
205	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
206	instantiation	288, 270, 217	⊢
	: , : , :
207	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
208	instantiation	221, 218	⊢
	: , : , :
209	instantiation	219, 220	⊢
	: , :
210	instantiation	221, 222	⊢
	: , : , :
211	theorem		⊢
	proveit.numbers.addition.add_complex_closure
212	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
213	instantiation	223	⊢
	: , : , :
214	instantiation	238, 224	⊢
	:
215	instantiation	238, 225	⊢
	:
216	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
217	instantiation	288, 277, 226	⊢
	: , : , :
218	instantiation	227, 245, 228	⊢
	: , :
219	theorem		⊢
	proveit.logic.equality.equals_reversal
220	instantiation	229, 261, 230, 254, 252	⊢
	: , : , :
221	axiom		⊢
	proveit.logic.equality.substitution
222	instantiation	231, 232, 233	⊢
	: , :
223	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
224	instantiation	234, 235, 236	⊢
	: , :
225	instantiation	288, 270, 237	⊢
	: , : , :
226	instantiation	288, 283, 281	⊢
	: , : , :
227	theorem		⊢
	proveit.numbers.addition.commutation
228	instantiation	238, 250	⊢
	:
229	theorem		⊢
	proveit.numbers.exponentiation.product_of_real_powers
230	instantiation	239, 258	⊢
	:
231	theorem		⊢
	proveit.numbers.exponentiation.neg_power_as_div
232	instantiation	288, 240, 241	⊢
	: , : , :
233	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
234	theorem		⊢
	proveit.numbers.multiplication.mult_complex_closure_bin
235	instantiation	242, 250, 243, 244	⊢
	: , :
236	instantiation	249, 261, 245	⊢
	: , :
237	instantiation	288, 277, 246	⊢
	: , : , :
238	theorem		⊢
	proveit.numbers.negation.complex_closure
239	theorem		⊢
	proveit.numbers.negation.real_closure
240	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
241	instantiation	288, 247, 248	⊢
	: , : , :
242	theorem		⊢
	proveit.numbers.division.div_complex_closure
243	instantiation	249, 261, 250	⊢
	: , :
244	instantiation	251, 252, 253	⊢
	: , : , :
245	instantiation	288, 270, 254	⊢
	: , : , :
246	instantiation	288, 283, 255	⊢
	: , : , :
247	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
248	instantiation	288, 256, 257	⊢
	: , : , :
249	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
250	instantiation	288, 270, 258	⊢
	: , : , :
251	theorem		⊢
	proveit.logic.equality.sub_left_side_into
252	instantiation	259, 268	⊢
	:
253	instantiation	260, 261	⊢
	:
254	instantiation	262, 263, 264	⊢
	: , : , :
255	instantiation	288, 265, 266	⊢
	: , : , :
256	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
257	instantiation	288, 267, 268	⊢
	: , : , :
258	instantiation	288, 277, 269	⊢
	: , : , :
259	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
260	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
261	instantiation	288, 270, 271	⊢
	: , : , :
262	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
263	instantiation	272, 273	⊢
	: , :
264	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
265	instantiation	274, 276, 275	⊢
	: , :
266	assumption		⊢
267	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
268	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
269	instantiation	288, 283, 276	⊢
	: , : , :
270	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
271	instantiation	288, 277, 278	⊢
	: , : , :
272	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
273	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
274	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
275	instantiation	279, 280, 281	⊢
	: , :
276	instantiation	288, 289, 282	⊢
	: , : , :
277	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
278	instantiation	288, 283, 287	⊢
	: , : , :
279	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
280	instantiation	288, 284, 285	⊢
	: , : , :
281	instantiation	286, 287	⊢
	:
282	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
283	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
284	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
285	theorem		⊢
	proveit.physics.quantum.QPE._two_pow_t_minus_one_is_nat_pos
286	theorem		⊢
	proveit.numbers.negation.int_closure
287	instantiation	288, 289, 290	⊢
	: , : , :
288	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
289	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
290	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
*equality replacement requirements