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	168	⊢
2	instantiation	4, 263, 34, 182, 24, 7, 184, 94, 8, 9, 5, 10	⊢
	: , : , : , : , : , : , : , : , : , :
3	instantiation	6, 11, 182, 34, 184, 24, 7, 94, 8, 9, 12, 10	⊢
	: , : , : , : , : , : , : , : , : , :
4	theorem		⊢
	proveit.linear_algebra.tensors.tensor_prod_disassociation
5	instantiation	93, 94, 11, 12	⊢
	: , : , : , :
6	theorem		⊢
	proveit.linear_algebra.tensors.factor_scalar_from_tensor_prod
7	instantiation	61, 13, 144, 16	⊢
	: , : , : , :
8	instantiation	14, 255, 256, 94, 27	⊢
	: , : , :
9	instantiation	61, 15, 144, 16	⊢
	: , : , : , :
10	modus ponens	17, 18	⊢
11	instantiation	97, 198, 19, 20	⊢
	: , :
12	instantiation	66, 94, 67, 21	⊢
	: , : , : , :
13	instantiation	158, 22, 24	⊢
	: , : , :
14	theorem		⊢
	proveit.logic.booleans.conjunction.redundant_conjunction_general
15	instantiation	158, 23, 24	⊢
	: , : , :
16	instantiation	88, 25	⊢
	: , :
17	instantiation	26, 255, 256, 27	⊢
	: , : , : , :
18	generalization	28	⊢
19	instantiation	209, 219, 29	⊢
	: , :
20	instantiation	30, 65, 31	⊢
	: , : , :
21	instantiation	93, 94, 32, 96	⊢
	: , : , : , :
22	instantiation	33, 34	⊢
	: , : , :
23	instantiation	33, 34	⊢
	: , : , :
24	instantiation	168, 35, 36	⊢
	: , : , :
25	instantiation	37, 208	⊢
	: , :
26	theorem		⊢
	proveit.logic.booleans.conjunction.conjunction_from_quantification
27	instantiation	126, 38, 39, 213, 40, 41*, 42*	⊢
	: , : , :
28	instantiation	93, 94, 43, 44	,  ⊢
	: , : , : , :
29	instantiation	196, 45, 46	⊢
	: , :
30	theorem		⊢
	proveit.logic.equality.sub_left_side_into
31	instantiation	186, 47	⊢
	: , : , :
32	instantiation	209, 114, 48	⊢
	: , :
33	theorem		⊢
	proveit.core_expr_types.tuples.range_len
34	instantiation	49, 162, 50, 182, 51, 263	⊢
	: , :
35	instantiation	186, 52	⊢
	: , : , :
36	instantiation	61, 53, 54, 55	⊢
	: , : , : , :
37	theorem		⊢
	proveit.core_expr_types.tuples.range_from1_len
38	instantiation	264, 239, 56	⊢
	: , : , :
39	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
40	instantiation	57, 58	⊢
	: , :
41	instantiation	168, 59, 60	⊢
	: , : , :
42	instantiation	61, 62, 82, 63	⊢
	: , : , : , :
43	instantiation	97, 198, 64, 65	⊢
	: , :
44	instantiation	66, 94, 67, 68	,  ⊢
	: , : , : , :
45	instantiation	97, 180, 219, 152	⊢
	: , :
46	instantiation	221, 69	⊢
	:
47	instantiation	168, 70, 71	⊢
	: , : , :
48	instantiation	158, 72, 73	⊢
	: , : , :
49	theorem		⊢
	proveit.numbers.addition.add_nat_closure
50	instantiation	179	⊢
	: , : , :
51	instantiation	74, 75	⊢
	:
52	instantiation	76, 142, 198, 77*	⊢
	: , :
53	instantiation	118, 263, 251, 78, 84, 197, 80, 198	⊢
	: , : , : , : , : , :
54	instantiation	85, 182, 162, 184, 79, 197, 80, 198	⊢
	: , : , : , :
55	instantiation	81, 198, 197, 82	⊢
	: , : , :
56	instantiation	264, 246, 255	⊢
	: , : , :
57	theorem		⊢
	proveit.numbers.ordering.relax_less
58	instantiation	83, 266	⊢
	:
59	instantiation	118, 263, 251, 182, 108, 184, 84, 142, 198	⊢
	: , : , : , : , : , :
60	instantiation	85, 182, 251, 184, 108, 142, 198	⊢
	: , : , : , :
61	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
62	instantiation	168, 86, 87	⊢
	: , : , :
63	instantiation	88, 89	⊢
	: , :
64	instantiation	90, 219	⊢
	:
65	instantiation	91, 228, 92	⊢
	: , :
66	theorem		⊢
	proveit.linear_algebra.addition.binary_closure
67	theorem		⊢
	proveit.physics.quantum.algebra.ket_zero_in_qubit_space
68	instantiation	93, 94, 95, 96	,  ⊢
	: , : , : , :
69	instantiation	97, 197, 219, 152	⊢
	: , :
70	instantiation	168, 98, 99	⊢
	: , : , :
71	instantiation	168, 100, 101	⊢
	: , : , :
72	instantiation	191, 161, 102	⊢
	: , :
73	instantiation	168, 103, 104	⊢
	: , : , :
74	theorem		⊢
	proveit.numbers.negation.nat_closure
75	instantiation	105, 255, 106	⊢
	:
76	theorem		⊢
	proveit.numbers.negation.distribute_neg_through_binary_sum
77	instantiation	107, 197	⊢
	:
78	instantiation	199	⊢
	: , :
79	instantiation	179	⊢
	: , : , :
80	instantiation	221, 198	⊢
	:
81	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
82	instantiation	157	⊢
	:
83	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
84	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.zero_is_complex
85	theorem		⊢
	proveit.numbers.addition.elim_zero_any
86	instantiation	118, 263, 251, 182, 108, 184, 197, 142, 198	⊢
	: , : , : , : , : , :
87	instantiation	109, 197, 198, 144	⊢
	: , : , :
88	theorem		⊢
	proveit.logic.equality.equals_reversal
89	instantiation	110, 198	⊢
	:
90	theorem		⊢
	proveit.numbers.exponentiation.sqrt_complex_closure
91	theorem		⊢
	proveit.numbers.exponentiation.exp_rational_non_zero__not_zero
92	instantiation	111, 112, 228	⊢
	: , :
93	theorem		⊢
	proveit.linear_algebra.scalar_multiplication.scalar_mult_closure
94	instantiation	113, 250	⊢
	:
95	instantiation	209, 114, 115	,  ⊢
	: , :
96	theorem		⊢
	proveit.physics.quantum.algebra.ket_one_in_qubit_space
97	theorem		⊢
	proveit.numbers.division.div_complex_closure
98	instantiation	186, 116	⊢
	: , : , :
99	instantiation	186, 117	⊢
	: , : , :
100	instantiation	118, 182, 251, 263, 184, 119, 136, 201, 120	⊢
	: , : , : , : , : , :
101	instantiation	121, 136, 201, 122	⊢
	: , : , :
102	instantiation	158, 123, 124	⊢
	: , : , :
103	instantiation	175, 263, 162, 182, 125, 184, 161, 192, 177, 139	⊢
	: , : , : , : , : , :
104	instantiation	175, 182, 251, 162, 184, 163, 125, 219, 178, 192, 177, 139	⊢
	: , : , : , : , : , :
105	theorem		⊢
	proveit.numbers.number_sets.integers.nonpos_int_is_int_nonpos
106	instantiation	126, 156, 214, 213, 127, 128*, 129*	⊢
	: , : , :
107	theorem		⊢
	proveit.numbers.negation.double_negation
108	instantiation	199	⊢
	: , :
109	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_12
110	theorem		⊢
	proveit.numbers.addition.elim_zero_left
111	theorem		⊢
	proveit.numbers.division.div_rational_nonzero_closure
112	instantiation	264, 236, 130	⊢
	: , : , :
113	theorem		⊢
	proveit.linear_algebra.complex_vec_set_is_vec_space
114	instantiation	264, 230, 131	⊢
	: , : , :
115	instantiation	158, 132, 133	,  ⊢
	: , : , :
116	instantiation	151, 180, 219, 152, 134*	⊢
	: , :
117	instantiation	186, 135	⊢
	: , : , :
118	theorem		⊢
	proveit.numbers.addition.disassociation
119	instantiation	199	⊢
	: , :
120	instantiation	221, 136	⊢
	:
121	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
122	instantiation	157	⊢
	:
123	instantiation	191, 137, 139	⊢
	: , :
124	instantiation	175, 182, 251, 263, 184, 138, 192, 177, 139	⊢
	: , : , : , : , : , :
125	instantiation	179	⊢
	: , : , :
126	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
127	instantiation	140, 266	⊢
	:
128	instantiation	141, 198, 142	⊢
	: , :
129	instantiation	143, 197, 144	⊢
	: , :
130	instantiation	264, 244, 248	⊢
	: , : , :
131	instantiation	264, 211, 145	⊢
	: , : , :
132	instantiation	191, 161, 146	,  ⊢
	: , :
133	instantiation	168, 147, 148	,  ⊢
	: , : , :
134	instantiation	168, 149, 150	⊢
	: , : , :
135	instantiation	151, 197, 219, 152, 153*	⊢
	: , :
136	instantiation	264, 230, 154	⊢
	: , : , :
137	instantiation	191, 192, 177	⊢
	: , :
138	instantiation	199	⊢
	: , :
139	instantiation	264, 230, 155	⊢
	: , : , :
140	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_lower_bound
141	theorem		⊢
	proveit.numbers.addition.commutation
142	instantiation	264, 230, 156	⊢
	: , : , :
143	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_basic
144	instantiation	157	⊢
	:
145	theorem		⊢
	proveit.numbers.number_sets.real_numbers.e_is_real_pos
146	instantiation	158, 159, 160	,  ⊢
	: , : , :
147	instantiation	175, 263, 162, 182, 164, 184, 161, 192, 193, 177	,  ⊢
	: , : , : , : , : , :
148	instantiation	175, 182, 251, 162, 184, 163, 164, 219, 178, 192, 193, 177	,  ⊢
	: , : , : , : , : , :
149	instantiation	186, 187	⊢
	: , : , :
150	instantiation	168, 165, 166	⊢
	: , : , :
151	theorem		⊢
	proveit.numbers.division.div_as_mult
152	instantiation	167, 250	⊢
	:
153	instantiation	168, 169, 170	⊢
	: , : , :
154	instantiation	264, 239, 171	⊢
	: , : , :
155	instantiation	264, 239, 172	⊢
	: , : , :
156	instantiation	264, 239, 173	⊢
	: , : , :
157	axiom		⊢
	proveit.logic.equality.equals_reflexivity
158	theorem		⊢
	proveit.logic.equality.sub_right_side_into
159	instantiation	191, 174, 177	,  ⊢
	: , :
160	instantiation	175, 182, 251, 263, 184, 176, 192, 193, 177	,  ⊢
	: , : , : , : , : , :
161	instantiation	191, 219, 178	⊢
	: , :
162	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
163	instantiation	199	⊢
	: , :
164	instantiation	179	⊢
	: , : , :
165	instantiation	188, 180, 201	⊢
	: , :
166	instantiation	181, 263, 251, 182, 183, 184, 201, 197, 198, 185*	⊢
	: , : , : , : , : , :
167	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
168	axiom		⊢
	proveit.logic.equality.equals_transitivity
169	instantiation	186, 187	⊢
	: , : , :
170	instantiation	188, 197, 201	⊢
	: , :
171	instantiation	264, 234, 189	⊢
	: , : , :
172	instantiation	264, 246, 190	⊢
	: , : , :
173	instantiation	264, 246, 258	⊢
	: , : , :
174	instantiation	191, 192, 193	,  ⊢
	: , :
175	theorem		⊢
	proveit.numbers.multiplication.disassociation
176	instantiation	199	⊢
	: , :
177	instantiation	264, 230, 194	⊢
	: , : , :
178	instantiation	264, 230, 195	⊢
	: , : , :
179	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
180	instantiation	196, 197, 198	⊢
	: , :
181	theorem		⊢
	proveit.numbers.multiplication.distribute_through_sum
182	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
183	instantiation	199	⊢
	: , :
184	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
185	instantiation	200, 201	⊢
	:
186	axiom		⊢
	proveit.logic.equality.substitution
187	instantiation	202, 203, 248, 204*	⊢
	: , :
188	theorem		⊢
	proveit.numbers.multiplication.commutation
189	instantiation	205, 235, 206	⊢
	: , :
190	instantiation	207, 245, 208	⊢
	: , :
191	theorem		⊢
	proveit.numbers.multiplication.mult_complex_closure_bin
192	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.i_is_complex
193	instantiation	209, 219, 210	,  ⊢
	: , :
194	theorem		⊢
	proveit.physics.quantum.QPE._phase_is_real
195	instantiation	264, 211, 212	⊢
	: , : , :
196	theorem		⊢
	proveit.numbers.addition.add_complex_closure_bin
197	instantiation	264, 230, 213	⊢
	: , : , :
198	instantiation	264, 230, 214	⊢
	: , : , :
199	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
200	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
201	instantiation	264, 230, 215	⊢
	: , : , :
202	theorem		⊢
	proveit.numbers.exponentiation.neg_power_as_div
203	instantiation	264, 216, 217	⊢
	: , : , :
204	instantiation	218, 219	⊢
	:
205	theorem		⊢
	proveit.numbers.multiplication.mult_rational_pos_closure_bin
206	instantiation	264, 249, 266	⊢
	: , : , :
207	theorem		⊢
	proveit.numbers.exponentiation.exp_int_closure
208	instantiation	264, 220, 266	⊢
	: , : , :
209	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
210	instantiation	221, 222	,  ⊢
	:
211	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
212	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pi_is_real_pos
213	instantiation	223, 224, 266	⊢
	: , : , :
214	instantiation	264, 239, 225	⊢
	: , : , :
215	instantiation	264, 239, 226	⊢
	: , : , :
216	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
217	instantiation	264, 227, 228	⊢
	: , : , :
218	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
219	instantiation	264, 230, 229	⊢
	: , : , :
220	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
221	theorem		⊢
	proveit.numbers.negation.complex_closure
222	instantiation	264, 230, 231	,  ⊢
	: , : , :
223	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
224	instantiation	232, 233	⊢
	: , :
225	instantiation	264, 246, 259	⊢
	: , : , :
226	instantiation	264, 234, 235	⊢
	: , : , :
227	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
228	instantiation	264, 236, 237	⊢
	: , : , :
229	instantiation	264, 239, 238	⊢
	: , : , :
230	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
231	instantiation	264, 239, 240	,  ⊢
	: , : , :
232	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
233	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
234	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
235	instantiation	241, 242, 243	⊢
	: , :
236	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
237	instantiation	264, 244, 250	⊢
	: , : , :
238	instantiation	264, 246, 245	⊢
	: , : , :
239	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
240	instantiation	264, 246, 247	,  ⊢
	: , : , :
241	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
242	instantiation	264, 249, 248	⊢
	: , : , :
243	instantiation	264, 249, 250	⊢
	: , : , :
244	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
245	instantiation	264, 262, 251	⊢
	: , : , :
246	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
247	instantiation	264, 252, 253	,  ⊢
	: , : , :
248	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
249	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
250	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
251	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
252	instantiation	254, 255, 256	⊢
	: , :
253	assumption		⊢
254	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
255	instantiation	257, 258, 259	⊢
	: , :
256	theorem		⊢
	proveit.numbers.number_sets.integers.zero_is_int
257	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
258	instantiation	260, 261	⊢
	:
259	instantiation	264, 262, 263	⊢
	: , : , :
260	theorem		⊢
	proveit.numbers.negation.int_closure
261	instantiation	264, 265, 266	⊢
	: , : , :
262	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
263	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
264	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
265	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
266	assumption		⊢
*equality replacement requirements