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	⊢
	: , : , :
1	theorem		⊢
	proveit.logic.sets.unification.union_inclusion
2	reference	221	⊢
3	instantiation	164	⊢
	: , :
4	instantiation	8, 57, 6, 235, 7	⊢
	: , : , : , :
5	instantiation	8, 57, 10, 235, 9	⊢
	: , : , : , :
6	instantiation	241, 10	⊢
	:
7	instantiation	15, 160, 11, 12, 13, 14	⊢
	: , :
8	theorem		⊢
	proveit.numbers.number_sets.integers.interval_subset_eq
9	instantiation	15, 160, 16, 17, 18, 19	⊢
	: , :
10	instantiation	234, 207, 229	⊢
	: , :
11	instantiation	171	⊢
	: , : , :
12	instantiation	32, 20, 21	⊢
	: , :
13	instantiation	44, 22, 186, 52, 53, 23*, 24*	⊢
	: , : , :
14	instantiation	49, 25	⊢
	: , :
15	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_all
16	instantiation	171	⊢
	: , : , :
17	instantiation	26, 27, 28	⊢
	: , : , :
18	instantiation	44, 213, 186, 29, 30, 31*	⊢
	: , : , :
19	instantiation	32, 45, 33	⊢
	: , :
20	instantiation	243, 230, 34	⊢
	: , : , :
21	instantiation	150	⊢
	:
22	instantiation	35, 160, 36, 83, 213, 143	⊢
	: , :
23	instantiation	155, 37, 38, 39	⊢
	: , : , : , :
24	instantiation	167, 40	⊢
	: , :
25	instantiation	41, 42, 43	⊢
	: , : , :
26	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
27	instantiation	44, 83, 213, 45, 46, 47*, 48*	⊢
	: , : , :
28	instantiation	49, 50	⊢
	: , :
29	instantiation	142, 52, 213	⊢
	: , :
30	instantiation	51, 186, 52, 213, 53, 54	⊢
	: , : , :
31	instantiation	155, 55, 137, 56	⊢
	: , : , : , :
32	theorem		⊢
	proveit.numbers.ordering.relax_equal_to_less_eq
33	instantiation	150	⊢
	:
34	instantiation	243, 238, 57	⊢
	: , : , :
35	theorem		⊢
	proveit.numbers.addition.add_real_closure
36	instantiation	171	⊢
	: , : , :
37	instantiation	132, 58, 59	⊢
	: , : , :
38	instantiation	150	⊢
	:
39	instantiation	167, 60	⊢
	: , :
40	instantiation	155, 61, 62, 63	⊢
	: , : , : , :
41	axiom		⊢
	proveit.numbers.ordering.transitivity_less_less
42	instantiation	64, 65	⊢
	:
43	instantiation	70, 240	⊢
	:
44	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
45	instantiation	196, 197, 240	⊢
	: , : , :
46	instantiation	66, 240	⊢
	:
47	instantiation	176, 202, 67	⊢
	: , :
48	instantiation	132, 68, 69	⊢
	: , : , :
49	theorem		⊢
	proveit.numbers.ordering.relax_less
50	instantiation	70, 82	⊢
	:
51	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right
52	instantiation	243, 230, 71	⊢
	: , : , :
53	instantiation	72, 229, 228, 219	⊢
	: , : , :
54	instantiation	73, 237	⊢
	:
55	instantiation	132, 74, 75	⊢
	: , : , :
56	instantiation	167, 144	⊢
	: , :
57	instantiation	234, 111, 229	⊢
	: , :
58	instantiation	169, 118	⊢
	: , : , :
59	instantiation	132, 76, 77	⊢
	: , : , :
60	instantiation	169, 131	⊢
	: , : , :
61	instantiation	78, 174, 202	⊢
	: , :
62	instantiation	167, 79	⊢
	: , :
63	instantiation	167, 80	⊢
	: , :
64	theorem		⊢
	proveit.numbers.number_sets.integers.negative_if_in_neg_int
65	instantiation	81, 82	⊢
	:
66	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_lower_bound
67	instantiation	243, 223, 83	⊢
	: , : , :
68	instantiation	132, 84, 85	⊢
	: , : , :
69	instantiation	86, 172, 137	⊢
	: , :
70	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
71	instantiation	243, 238, 228	⊢
	: , : , :
72	theorem		⊢
	proveit.numbers.number_sets.integers.interval_upper_bound
73	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_lower_bound
74	instantiation	169, 87	⊢
	: , : , :
75	instantiation	132, 88, 89	⊢
	: , : , :
76	instantiation	146, 237, 160, 147, 161, 149, 174, 162, 202, 163	⊢
	: , : , : , : , : , :
77	instantiation	135, 147, 245, 149, 90, 174, 162, 202, 129	⊢
	: , : , : , : , : , : , : , :
78	theorem		⊢
	proveit.numbers.negation.distribute_neg_through_binary_sum
79	instantiation	91, 116, 174, 178, 92	⊢
	: , : , :
80	instantiation	132, 93, 94	⊢
	: , : , :
81	theorem		⊢
	proveit.numbers.negation.int_neg_closure
82	instantiation	95, 96, 182	⊢
	: , :
83	instantiation	243, 230, 97	⊢
	: , : , :
84	instantiation	169, 144	⊢
	: , : , :
85	instantiation	169, 131	⊢
	: , : , :
86	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_basic
87	instantiation	132, 98, 99	⊢
	: , : , :
88	instantiation	146, 147, 245, 237, 149, 100, 172, 178, 202	⊢
	: , : , : , : , : , :
89	instantiation	101, 202, 172, 102	⊢
	: , : , :
90	instantiation	164	⊢
	: , :
91	theorem		⊢
	proveit.numbers.addition.subtraction.subtract_from_add
92	instantiation	139, 103, 104	⊢
	: , : , :
93	instantiation	132, 105, 106	⊢
	: , : , :
94	instantiation	132, 107, 108	⊢
	: , : , :
95	theorem		⊢
	proveit.numbers.addition.add_nat_pos_closure_bin
96	instantiation	109, 207, 110	⊢
	:
97	instantiation	243, 238, 111	⊢
	: , : , :
98	instantiation	169, 117	⊢
	: , : , :
99	instantiation	132, 112, 113	⊢
	: , : , :
100	instantiation	164	⊢
	: , :
101	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
102	instantiation	150	⊢
	:
103	instantiation	132, 114, 115	⊢
	: , : , :
104	instantiation	176, 174, 116	⊢
	: , :
105	instantiation	169, 117	⊢
	: , : , :
106	instantiation	169, 118	⊢
	: , : , :
107	instantiation	132, 119, 120	⊢
	: , : , :
108	instantiation	124, 147, 245, 237, 149, 125, 153, 202, 163, 126*	⊢
	: , : , : , : , : , :
109	theorem		⊢
	proveit.numbers.number_sets.integers.pos_int_is_natural_pos
110	instantiation	121, 122, 123	⊢
	: , : , :
111	instantiation	241, 235	⊢
	:
112	instantiation	146, 147, 245, 237, 149, 148, 172, 153, 202	⊢
	: , : , : , : , : , :
113	instantiation	124, 237, 245, 147, 125, 149, 172, 153, 202, 126*	⊢
	: , : , : , : , : , :
114	instantiation	146, 237, 245, 147, 127, 149, 174, 163, 178	⊢
	: , : , : , : , : , :
115	instantiation	128, 174, 178, 129	⊢
	: , : , :
116	instantiation	243, 223, 130	⊢
	: , : , :
117	instantiation	169, 144	⊢
	: , : , :
118	instantiation	169, 131	⊢
	: , : , :
119	instantiation	132, 133, 134	⊢
	: , : , :
120	instantiation	135, 147, 237, 245, 149, 136, 172, 153, 202, 163, 137	⊢
	: , : , : , : , : , : , : , :
121	theorem		⊢
	proveit.numbers.ordering.transitivity_less_less_eq
122	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
123	instantiation	138, 229, 228, 219	⊢
	: , : , :
124	theorem		⊢
	proveit.numbers.addition.association
125	instantiation	164	⊢
	: , :
126	instantiation	139, 140, 141	⊢
	: , : , :
127	instantiation	164	⊢
	: , :
128	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_12
129	instantiation	150	⊢
	:
130	instantiation	142, 143, 188	⊢
	: , :
131	instantiation	169, 144	⊢
	: , : , :
132	axiom		⊢
	proveit.logic.equality.equals_transitivity
133	instantiation	146, 147, 245, 237, 149, 148, 172, 153, 145	⊢
	: , : , : , : , : , :
134	instantiation	146, 245, 160, 147, 148, 161, 149, 172, 153, 162, 202, 163	⊢
	: , : , : , : , : , :
135	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_general
136	instantiation	164	⊢
	: , :
137	instantiation	150	⊢
	:
138	theorem		⊢
	proveit.numbers.number_sets.integers.interval_lower_bound
139	theorem		⊢
	proveit.logic.equality.sub_right_side_into
140	instantiation	151, 202, 216, 152	⊢
	: , : , :
141	instantiation	176, 202, 153	⊢
	: , :
142	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
143	instantiation	154, 186	⊢
	:
144	instantiation	155, 156, 157, 158	⊢
	: , : , : , :
145	instantiation	159, 160, 161, 162, 202, 163	⊢
	: , :
146	theorem		⊢
	proveit.numbers.addition.disassociation
147	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
148	instantiation	164	⊢
	: , :
149	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
150	axiom		⊢
	proveit.logic.equality.equals_reflexivity
151	theorem		⊢
	proveit.numbers.addition.subtraction.subtract_from_add_reversed
152	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
153	instantiation	243, 223, 165	⊢
	: , : , :
154	theorem		⊢
	proveit.numbers.negation.real_closure
155	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
156	instantiation	169, 166	⊢
	: , : , :
157	instantiation	167, 168	⊢
	: , :
158	instantiation	169, 170	⊢
	: , : , :
159	theorem		⊢
	proveit.numbers.addition.add_complex_closure
160	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
161	instantiation	171	⊢
	: , : , :
162	instantiation	173, 172	⊢
	:
163	instantiation	173, 174	⊢
	:
164	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
165	instantiation	243, 230, 175	⊢
	: , : , :
166	instantiation	176, 177, 178	⊢
	: , :
167	theorem		⊢
	proveit.logic.equality.equals_reversal
168	instantiation	179, 216, 188, 187, 204	⊢
	: , : , :
169	axiom		⊢
	proveit.logic.equality.substitution
170	instantiation	180, 181, 182	⊢
	: , :
171	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
172	instantiation	183, 184, 185	⊢
	: , :
173	theorem		⊢
	proveit.numbers.negation.complex_closure
174	instantiation	243, 223, 186	⊢
	: , : , :
175	instantiation	243, 238, 236	⊢
	: , : , :
176	theorem		⊢
	proveit.numbers.addition.commutation
177	instantiation	243, 223, 187	⊢
	: , : , :
178	instantiation	243, 223, 188	⊢
	: , : , :
179	theorem		⊢
	proveit.numbers.exponentiation.product_of_real_powers
180	theorem		⊢
	proveit.numbers.exponentiation.neg_power_as_div
181	instantiation	243, 189, 190	⊢
	: , : , :
182	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
183	theorem		⊢
	proveit.numbers.multiplication.mult_complex_closure_bin
184	instantiation	191, 202, 192, 193	⊢
	: , :
185	instantiation	243, 223, 194	⊢
	: , : , :
186	instantiation	243, 230, 195	⊢
	: , : , :
187	instantiation	196, 197, 233	⊢
	: , : , :
188	instantiation	243, 230, 198	⊢
	: , : , :
189	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
190	instantiation	243, 199, 200	⊢
	: , : , :
191	theorem		⊢
	proveit.numbers.division.div_complex_closure
192	instantiation	201, 216, 202	⊢
	: , :
193	instantiation	203, 204, 205	⊢
	: , : , :
194	instantiation	243, 230, 206	⊢
	: , : , :
195	instantiation	243, 238, 207	⊢
	: , : , :
196	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
197	instantiation	208, 209	⊢
	: , :
198	instantiation	243, 238, 210	⊢
	: , : , :
199	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
200	instantiation	243, 211, 212	⊢
	: , : , :
201	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
202	instantiation	243, 223, 213	⊢
	: , : , :
203	theorem		⊢
	proveit.logic.equality.sub_left_side_into
204	instantiation	214, 221	⊢
	:
205	instantiation	215, 216	⊢
	:
206	instantiation	243, 238, 217	⊢
	: , : , :
207	instantiation	243, 218, 219	⊢
	: , : , :
208	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
209	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
210	instantiation	241, 229	⊢
	:
211	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
212	instantiation	243, 220, 221	⊢
	: , : , :
213	instantiation	243, 230, 222	⊢
	: , : , :
214	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
215	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
216	instantiation	243, 223, 224	⊢
	: , : , :
217	instantiation	225, 242, 226	⊢
	: , :
218	instantiation	227, 229, 228	⊢
	: , :
219	assumption		⊢
220	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
221	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
222	instantiation	243, 238, 229	⊢
	: , : , :
223	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
224	instantiation	243, 230, 231	⊢
	: , : , :
225	theorem		⊢
	proveit.numbers.exponentiation.exp_int_closure
226	instantiation	243, 232, 233	⊢
	: , : , :
227	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
228	instantiation	234, 235, 236	⊢
	: , :
229	instantiation	243, 244, 237	⊢
	: , : , :
230	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
231	instantiation	243, 238, 242	⊢
	: , : , :
232	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
233	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
234	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
235	instantiation	243, 239, 240	⊢
	: , : , :
236	instantiation	241, 242	⊢
	:
237	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
238	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
239	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
240	theorem		⊢
	proveit.physics.quantum.QPE._two_pow_t_minus_one_is_nat_pos
241	theorem		⊢
	proveit.numbers.negation.int_closure
242	instantiation	243, 244, 245	⊢
	: , : , :
243	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
244	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
245	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
*equality replacement requirements