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, 8, 9, 10, 11, 12	⊢
	: , : , : , : , : , : , : , : , : , :
1	theorem		⊢
	proveit.linear_algebra.tensors.tensor_prod_disassociation
2	reference	265	⊢
3	reference	32	⊢
4	reference	184	⊢
5	reference	23	⊢
6	instantiation	63, 13, 140, 16	⊢
	: , : , : , :
7	reference	186	⊢
8	reference	96	⊢
9	instantiation	14, 257, 258, 96, 29	⊢
	: , : , :
10	instantiation	63, 15, 140, 16	⊢
	: , : , : , :
11	instantiation	95, 96, 17, 18	⊢
	: , : , : , :
12	modus ponens	19, 20	⊢
13	instantiation	160, 21, 23	⊢
	: , : , :
14	theorem		⊢
	proveit.logic.booleans.conjunction.redundant_conjunction_general
15	instantiation	160, 22, 23	⊢
	: , : , :
16	instantiation	90, 24	⊢
	: , :
17	instantiation	102, 200, 25, 26	⊢
	: , :
18	instantiation	68, 96, 69, 27	⊢
	: , : , : , :
19	instantiation	28, 257, 258, 29	⊢
	: , : , : , :
20	generalization	30	⊢
21	instantiation	31, 32	⊢
	: , : , :
22	instantiation	31, 32	⊢
	: , : , :
23	instantiation	171, 33, 34	⊢
	: , : , :
24	instantiation	35, 210	⊢
	: , :
25	instantiation	211, 221, 36	⊢
	: , :
26	instantiation	37, 67, 38	⊢
	: , : , :
27	instantiation	95, 96, 39, 98	⊢
	: , : , : , :
28	theorem		⊢
	proveit.logic.booleans.conjunction.conjunction_from_quantification
29	instantiation	118, 40, 41, 215, 42, 43*, 44*	⊢
	: , : , :
30	instantiation	95, 96, 45, 46	,  ⊢
	: , : , : , :
31	theorem		⊢
	proveit.core_expr_types.tuples.range_len
32	instantiation	47, 164, 48, 184, 49, 265	⊢
	: , :
33	instantiation	188, 50	⊢
	: , : , :
34	instantiation	63, 51, 52, 53	⊢
	: , : , : , :
35	theorem		⊢
	proveit.core_expr_types.tuples.range_from1_len
36	instantiation	198, 54, 55	⊢
	: , :
37	theorem		⊢
	proveit.logic.equality.sub_left_side_into
38	instantiation	188, 56	⊢
	: , : , :
39	instantiation	211, 116, 57	⊢
	: , :
40	instantiation	266, 241, 58	⊢
	: , : , :
41	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
42	instantiation	59, 60	⊢
	: , :
43	instantiation	171, 61, 62	⊢
	: , : , :
44	instantiation	63, 64, 79, 65	⊢
	: , : , : , :
45	instantiation	102, 200, 66, 67	⊢
	: , :
46	instantiation	68, 96, 69, 70	,  ⊢
	: , : , : , :
47	theorem		⊢
	proveit.numbers.addition.add_nat_closure
48	instantiation	181	⊢
	: , : , :
49	instantiation	71, 72	⊢
	:
50	instantiation	73, 138, 200, 74*	⊢
	: , :
51	instantiation	124, 265, 253, 75, 86, 199, 77, 200	⊢
	: , : , : , : , : , :
52	instantiation	87, 184, 164, 186, 76, 199, 77, 200	⊢
	: , : , : , :
53	instantiation	78, 200, 199, 79	⊢
	: , : , :
54	instantiation	102, 182, 221, 156	⊢
	: , :
55	instantiation	223, 80	⊢
	:
56	instantiation	171, 81, 82	⊢
	: , : , :
57	instantiation	160, 83, 84	⊢
	: , : , :
58	instantiation	266, 248, 257	⊢
	: , : , :
59	theorem		⊢
	proveit.numbers.ordering.relax_less
60	instantiation	85, 268	⊢
	:
61	instantiation	124, 265, 253, 184, 110, 186, 86, 138, 200	⊢
	: , : , : , : , : , :
62	instantiation	87, 184, 253, 186, 110, 138, 200	⊢
	: , : , : , :
63	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
64	instantiation	171, 88, 89	⊢
	: , : , :
65	instantiation	90, 91	⊢
	: , :
66	instantiation	92, 221	⊢
	:
67	instantiation	93, 230, 94	⊢
	: , :
68	theorem		⊢
	proveit.linear_algebra.addition.binary_closure
69	theorem		⊢
	proveit.physics.quantum.algebra.ket_zero_in_qubit_space
70	instantiation	95, 96, 97, 98	,  ⊢
	: , : , : , :
71	theorem		⊢
	proveit.numbers.negation.nat_closure
72	instantiation	99, 257, 100	⊢
	:
73	theorem		⊢
	proveit.numbers.negation.distribute_neg_through_binary_sum
74	instantiation	101, 199	⊢
	:
75	instantiation	201	⊢
	: , :
76	instantiation	181	⊢
	: , : , :
77	instantiation	223, 200	⊢
	:
78	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
79	instantiation	152	⊢
	:
80	instantiation	102, 199, 221, 156	⊢
	: , :
81	instantiation	171, 103, 104	⊢
	: , : , :
82	instantiation	171, 105, 106	⊢
	: , : , :
83	instantiation	193, 163, 107	⊢
	: , :
84	instantiation	171, 108, 109	⊢
	: , : , :
85	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
86	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.zero_is_complex
87	theorem		⊢
	proveit.numbers.addition.elim_zero_any
88	instantiation	124, 265, 253, 184, 110, 186, 199, 138, 200	⊢
	: , : , : , : , : , :
89	instantiation	111, 199, 200, 140	⊢
	: , : , :
90	theorem		⊢
	proveit.logic.equality.equals_reversal
91	instantiation	112, 200	⊢
	:
92	theorem		⊢
	proveit.numbers.exponentiation.sqrt_complex_closure
93	theorem		⊢
	proveit.numbers.exponentiation.exp_rational_non_zero__not_zero
94	instantiation	113, 114, 230	⊢
	: , :
95	theorem		⊢
	proveit.linear_algebra.scalar_multiplication.scalar_mult_closure
96	instantiation	115, 252	⊢
	:
97	instantiation	211, 116, 117	,  ⊢
	: , :
98	theorem		⊢
	proveit.physics.quantum.algebra.ket_one_in_qubit_space
99	theorem		⊢
	proveit.numbers.number_sets.integers.nonpos_int_is_int_nonpos
100	instantiation	118, 151, 216, 215, 119, 120*, 121*	⊢
	: , : , :
101	theorem		⊢
	proveit.numbers.negation.double_negation
102	theorem		⊢
	proveit.numbers.division.div_complex_closure
103	instantiation	188, 122	⊢
	: , : , :
104	instantiation	188, 123	⊢
	: , : , :
105	instantiation	124, 184, 253, 265, 186, 125, 143, 203, 126	⊢
	: , : , : , : , : , :
106	instantiation	127, 143, 203, 128	⊢
	: , : , :
107	instantiation	160, 129, 130	⊢
	: , : , :
108	instantiation	177, 265, 164, 184, 131, 186, 163, 194, 179, 146	⊢
	: , : , : , : , : , :
109	instantiation	177, 184, 253, 164, 186, 165, 131, 221, 180, 194, 179, 146	⊢
	: , : , : , : , : , :
110	instantiation	201	⊢
	: , :
111	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_12
112	theorem		⊢
	proveit.numbers.addition.elim_zero_left
113	theorem		⊢
	proveit.numbers.division.div_rational_nonzero_closure
114	instantiation	266, 238, 132	⊢
	: , : , :
115	theorem		⊢
	proveit.linear_algebra.complex_vec_set_is_vec_space
116	instantiation	266, 232, 133	⊢
	: , : , :
117	instantiation	160, 134, 135	,  ⊢
	: , : , :
118	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
119	instantiation	136, 268	⊢
	:
120	instantiation	137, 200, 138	⊢
	: , :
121	instantiation	139, 199, 140	⊢
	: , :
122	instantiation	155, 182, 221, 156, 141*	⊢
	: , :
123	instantiation	188, 142	⊢
	: , : , :
124	theorem		⊢
	proveit.numbers.addition.disassociation
125	instantiation	201	⊢
	: , :
126	instantiation	223, 143	⊢
	:
127	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
128	instantiation	152	⊢
	:
129	instantiation	193, 144, 146	⊢
	: , :
130	instantiation	177, 184, 253, 265, 186, 145, 194, 179, 146	⊢
	: , : , : , : , : , :
131	instantiation	181	⊢
	: , : , :
132	instantiation	266, 246, 250	⊢
	: , : , :
133	instantiation	266, 213, 147	⊢
	: , : , :
134	instantiation	193, 163, 148	,  ⊢
	: , :
135	instantiation	171, 149, 150	,  ⊢
	: , : , :
136	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_lower_bound
137	theorem		⊢
	proveit.numbers.addition.commutation
138	instantiation	266, 232, 151	⊢
	: , : , :
139	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_basic
140	instantiation	152	⊢
	:
141	instantiation	171, 153, 154	⊢
	: , : , :
142	instantiation	155, 199, 221, 156, 157*	⊢
	: , :
143	instantiation	266, 232, 158	⊢
	: , : , :
144	instantiation	193, 194, 179	⊢
	: , :
145	instantiation	201	⊢
	: , :
146	instantiation	266, 232, 159	⊢
	: , : , :
147	theorem		⊢
	proveit.numbers.number_sets.real_numbers.e_is_real_pos
148	instantiation	160, 161, 162	,  ⊢
	: , : , :
149	instantiation	177, 265, 164, 184, 166, 186, 163, 194, 195, 179	,  ⊢
	: , : , : , : , : , :
150	instantiation	177, 184, 253, 164, 186, 165, 166, 221, 180, 194, 195, 179	,  ⊢
	: , : , : , : , : , :
151	instantiation	266, 241, 167	⊢
	: , : , :
152	axiom		⊢
	proveit.logic.equality.equals_reflexivity
153	instantiation	188, 189	⊢
	: , : , :
154	instantiation	171, 168, 169	⊢
	: , : , :
155	theorem		⊢
	proveit.numbers.division.div_as_mult
156	instantiation	170, 252	⊢
	:
157	instantiation	171, 172, 173	⊢
	: , : , :
158	instantiation	266, 241, 174	⊢
	: , : , :
159	instantiation	266, 241, 175	⊢
	: , : , :
160	theorem		⊢
	proveit.logic.equality.sub_right_side_into
161	instantiation	193, 176, 179	,  ⊢
	: , :
162	instantiation	177, 184, 253, 265, 186, 178, 194, 195, 179	,  ⊢
	: , : , : , : , : , :
163	instantiation	193, 221, 180	⊢
	: , :
164	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
165	instantiation	201	⊢
	: , :
166	instantiation	181	⊢
	: , : , :
167	instantiation	266, 248, 260	⊢
	: , : , :
168	instantiation	190, 182, 203	⊢
	: , :
169	instantiation	183, 265, 253, 184, 185, 186, 203, 199, 200, 187*	⊢
	: , : , : , : , : , :
170	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
171	axiom		⊢
	proveit.logic.equality.equals_transitivity
172	instantiation	188, 189	⊢
	: , : , :
173	instantiation	190, 199, 203	⊢
	: , :
174	instantiation	266, 236, 191	⊢
	: , : , :
175	instantiation	266, 248, 192	⊢
	: , : , :
176	instantiation	193, 194, 195	,  ⊢
	: , :
177	theorem		⊢
	proveit.numbers.multiplication.disassociation
178	instantiation	201	⊢
	: , :
179	instantiation	266, 232, 196	⊢
	: , : , :
180	instantiation	266, 232, 197	⊢
	: , : , :
181	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
182	instantiation	198, 199, 200	⊢
	: , :
183	theorem		⊢
	proveit.numbers.multiplication.distribute_through_sum
184	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
185	instantiation	201	⊢
	: , :
186	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
187	instantiation	202, 203	⊢
	:
188	axiom		⊢
	proveit.logic.equality.substitution
189	instantiation	204, 205, 250, 206*	⊢
	: , :
190	theorem		⊢
	proveit.numbers.multiplication.commutation
191	instantiation	207, 237, 208	⊢
	: , :
192	instantiation	209, 247, 210	⊢
	: , :
193	theorem		⊢
	proveit.numbers.multiplication.mult_complex_closure_bin
194	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.i_is_complex
195	instantiation	211, 221, 212	,  ⊢
	: , :
196	theorem		⊢
	proveit.physics.quantum.QPE._phase_is_real
197	instantiation	266, 213, 214	⊢
	: , : , :
198	theorem		⊢
	proveit.numbers.addition.add_complex_closure_bin
199	instantiation	266, 232, 215	⊢
	: , : , :
200	instantiation	266, 232, 216	⊢
	: , : , :
201	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
202	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
203	instantiation	266, 232, 217	⊢
	: , : , :
204	theorem		⊢
	proveit.numbers.exponentiation.neg_power_as_div
205	instantiation	266, 218, 219	⊢
	: , : , :
206	instantiation	220, 221	⊢
	:
207	theorem		⊢
	proveit.numbers.multiplication.mult_rational_pos_closure_bin
208	instantiation	266, 251, 268	⊢
	: , : , :
209	theorem		⊢
	proveit.numbers.exponentiation.exp_int_closure
210	instantiation	266, 222, 268	⊢
	: , : , :
211	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
212	instantiation	223, 224	,  ⊢
	:
213	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
214	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pi_is_real_pos
215	instantiation	225, 226, 268	⊢
	: , : , :
216	instantiation	266, 241, 227	⊢
	: , : , :
217	instantiation	266, 241, 228	⊢
	: , : , :
218	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
219	instantiation	266, 229, 230	⊢
	: , : , :
220	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
221	instantiation	266, 232, 231	⊢
	: , : , :
222	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
223	theorem		⊢
	proveit.numbers.negation.complex_closure
224	instantiation	266, 232, 233	,  ⊢
	: , : , :
225	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
226	instantiation	234, 235	⊢
	: , :
227	instantiation	266, 248, 261	⊢
	: , : , :
228	instantiation	266, 236, 237	⊢
	: , : , :
229	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
230	instantiation	266, 238, 239	⊢
	: , : , :
231	instantiation	266, 241, 240	⊢
	: , : , :
232	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
233	instantiation	266, 241, 242	,  ⊢
	: , : , :
234	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
235	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
236	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
237	instantiation	243, 244, 245	⊢
	: , :
238	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
239	instantiation	266, 246, 252	⊢
	: , : , :
240	instantiation	266, 248, 247	⊢
	: , : , :
241	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
242	instantiation	266, 248, 249	,  ⊢
	: , : , :
243	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
244	instantiation	266, 251, 250	⊢
	: , : , :
245	instantiation	266, 251, 252	⊢
	: , : , :
246	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
247	instantiation	266, 264, 253	⊢
	: , : , :
248	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
249	instantiation	266, 254, 255	,  ⊢
	: , : , :
250	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
251	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
252	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
253	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
254	instantiation	256, 257, 258	⊢
	: , :
255	assumption		⊢
256	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
257	instantiation	259, 260, 261	⊢
	: , :
258	theorem		⊢
	proveit.numbers.number_sets.integers.zero_is_int
259	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
260	instantiation	262, 263	⊢
	:
261	instantiation	266, 264, 265	⊢
	: , : , :
262	theorem		⊢
	proveit.numbers.negation.int_closure
263	instantiation	266, 267, 268	⊢
	: , : , :
264	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
265	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
266	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
267	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
268	assumption		⊢
*equality replacement requirements