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	generalization	1	⊢
1	instantiation	2, 3, 4	⊢
	: , :
2	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
3	instantiation	190, 5, 14	⊢
	: , : , :
4	instantiation	6, 328, 20, 7, 22, 23, 8, 9*	⊢
	: , : , : , :
5	instantiation	244, 10, 11	⊢
	: , : , :
6	theorem		⊢
	proveit.linear_algebra.tensors.norm_preserving_tensor_prod
7	instantiation	31, 317, 318, 134, 46	⊢
	: , : , :
8	modus ponens	12, 13	⊢
9	instantiation	211, 14	⊢
	: , :
10	instantiation	15, 328, 16, 17, 18*	⊢
	: , : , :
11	instantiation	19, 328, 20, 21, 22, 23	⊢
	: , : , : , :
12	instantiation	45, 317, 318, 46	⊢
	: , : , : , :
13	generalization	24	⊢
14	instantiation	25, 328	⊢
	:
15	theorem		⊢
	proveit.linear_algebra.tensors.tensor_prod_of_cart_exps_within_cart_exp
16	instantiation	107, 26, 204, 33	⊢
	: , : , : , :
17	instantiation	31, 317, 318, 260, 46	⊢
	: , : , :
18	instantiation	27, 28, 297, 43, 29*, 43*	⊢
	: , :
19	theorem		⊢
	proveit.linear_algebra.tensors.tensor_prod_is_in_tensor_prod_space
20	instantiation	107, 30, 204, 33	⊢
	: , : , : , :
21	instantiation	31, 317, 318, 130, 46	⊢
	: , : , :
22	instantiation	107, 32, 204, 33	⊢
	: , : , : , :
23	modus ponens	34, 35	⊢
24	instantiation	253, 36, 37	,  ⊢
	: , : , :
25	axiom		⊢
	proveit.physics.quantum.QPE._psi_t_def
26	instantiation	261, 38, 43	⊢
	: , : , :
27	theorem		⊢
	proveit.numbers.exponentiation.exp_nat_pos_rev
28	instantiation	149, 311, 278, 39, 279, 83, 325, 150	⊢
	: , : , : , : , :
29	instantiation	211, 40	⊢
	: , :
30	instantiation	261, 41, 43	⊢
	: , : , :
31	theorem		⊢
	proveit.logic.booleans.conjunction.redundant_conjunction_general
32	instantiation	261, 42, 43	⊢
	: , : , :
33	instantiation	211, 44	⊢
	: , :
34	instantiation	45, 317, 318, 46	⊢
	: , : , : , :
35	generalization	47	⊢
36	instantiation	102, 134, 67, 68, 48*, 49*	,  ⊢
	: , : , : , :
37	instantiation	253, 50, 51	⊢
	: , : , :
38	instantiation	56, 57	⊢
	: , : , :
39	instantiation	290	⊢
	: , :
40	instantiation	52, 320, 53, 54, 55	⊢
	: , : , : , : , : , :
41	instantiation	56, 57	⊢
	: , : , :
42	instantiation	56, 57	⊢
	: , : , :
43	instantiation	253, 58, 59	⊢
	: , : , :
44	instantiation	60, 61	⊢
	: , :
45	theorem		⊢
	proveit.logic.booleans.conjunction.conjunction_from_quantification
46	instantiation	178, 62, 63, 223, 64, 65*, 66*	⊢
	: , : , :
47	instantiation	129, 130, 67, 68	,  ⊢
	: , : , : , :
48	instantiation	69, 70	⊢
	:
49	instantiation	71, 130, 135, 97, 72, 73*	,  ⊢
	: , : , : , :
50	instantiation	235, 74	⊢
	: , : , :
51	instantiation	261, 75, 76	⊢
	: , : , :
52	theorem		⊢
	proveit.core_expr_types.tuples.shift_equivalence
53	instantiation	77, 78, 83	⊢
	: , :
54	instantiation	211, 79	⊢
	: , :
55	instantiation	211, 80	⊢
	: , :
56	theorem		⊢
	proveit.core_expr_types.tuples.range_len
57	instantiation	81, 265, 82, 278, 83, 325	⊢
	: , :
58	instantiation	235, 84	⊢
	: , : , :
59	instantiation	107, 85, 86, 87	⊢
	: , : , : , :
60	theorem		⊢
	proveit.core_expr_types.tuples.range_from1_len
61	instantiation	326, 111, 328	⊢
	: , : , :
62	instantiation	326, 309, 88	⊢
	: , : , :
63	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
64	instantiation	89, 90	⊢
	: , :
65	instantiation	253, 91, 92	⊢
	: , : , :
66	instantiation	107, 93, 123, 94	⊢
	: , : , : , :
67	instantiation	326, 306, 95	⊢
	: , : , :
68	instantiation	96, 130, 135, 97	,  ⊢
	: , : , : , :
69	theorem		⊢
	proveit.numbers.absolute_value.abs_non_neg_elim
70	instantiation	98, 99	⊢
	:
71	theorem		⊢
	proveit.linear_algebra.addition.norm_of_sum_of_orthogonal_pair
72	instantiation	261, 100, 101	,  ⊢
	: , : , :
73	instantiation	102, 134, 186, 136, 103*	,  ⊢
	: , : , : , :
74	instantiation	235, 104	⊢
	: , : , :
75	instantiation	261, 105, 106	⊢
	: , : , :
76	instantiation	107, 108, 109, 110	⊢
	: , : , : , :
77	theorem		⊢
	proveit.numbers.addition.add_nat_closure_bin
78	instantiation	326, 111, 112	⊢
	: , : , :
79	instantiation	253, 113, 114	⊢
	: , : , :
80	instantiation	253, 115, 116	⊢
	: , : , :
81	theorem		⊢
	proveit.numbers.addition.add_nat_closure
82	instantiation	283	⊢
	: , : , :
83	instantiation	284, 117	⊢
	:
84	instantiation	118, 201, 200, 153*	⊢
	: , :
85	instantiation	174, 325, 311, 119, 164, 203, 121, 200	⊢
	: , : , : , : , : , :
86	instantiation	125, 278, 265, 279, 120, 203, 121, 200	⊢
	: , : , : , :
87	instantiation	122, 200, 203, 123	⊢
	: , : , :
88	instantiation	326, 312, 317	⊢
	: , : , :
89	theorem		⊢
	proveit.numbers.ordering.relax_less
90	instantiation	124, 328	⊢
	:
91	instantiation	174, 325, 311, 278, 175, 279, 164, 201, 200	⊢
	: , : , : , : , : , :
92	instantiation	125, 278, 311, 279, 175, 201, 200	⊢
	: , : , : , :
93	instantiation	253, 126, 127	⊢
	: , : , :
94	instantiation	211, 128	⊢
	: , :
95	instantiation	326, 299, 132	⊢
	: , : , :
96	theorem		⊢
	proveit.linear_algebra.addition.binary_closure
97	instantiation	129, 130, 186, 136	,  ⊢
	: , : , : , :
98	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonneg_if_in_real_nonneg
99	instantiation	326, 131, 132	⊢
	: , : , :
100	instantiation	133, 134, 186, 135, 136	,  ⊢
	: , : , : , : , :
101	instantiation	253, 137, 138	,  ⊢
	: , : , :
102	theorem		⊢
	proveit.linear_algebra.inner_products.scaled_norm
103	instantiation	253, 139, 140	,  ⊢
	: , : , :
104	instantiation	253, 141, 142	⊢
	: , : , :
105	instantiation	143, 200, 144, 145	⊢
	: , : , : , : , :
106	instantiation	253, 146, 147	⊢
	: , : , :
107	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
108	instantiation	235, 148	⊢
	: , : , :
109	instantiation	235, 148	⊢
	: , : , :
110	instantiation	240, 200	⊢
	:
111	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
112	instantiation	149, 325, 278, 279, 150	⊢
	: , : , : , : , :
113	instantiation	235, 153	⊢
	: , : , :
114	instantiation	253, 151, 152	⊢
	: , : , :
115	instantiation	235, 153	⊢
	: , : , :
116	instantiation	156, 203	⊢
	:
117	instantiation	294, 317, 154	⊢
	:
118	theorem		⊢
	proveit.numbers.negation.distribute_neg_through_binary_sum
119	instantiation	290	⊢
	: , :
120	instantiation	283	⊢
	: , : , :
121	instantiation	303, 200	⊢
	:
122	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
123	instantiation	224	⊢
	:
124	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
125	theorem		⊢
	proveit.numbers.addition.elim_zero_any
126	instantiation	174, 325, 311, 278, 175, 279, 203, 201, 200	⊢
	: , : , : , : , : , :
127	instantiation	155, 203, 200, 204	⊢
	: , : , :
128	instantiation	156, 200	⊢
	:
129	theorem		⊢
	proveit.linear_algebra.scalar_multiplication.scalar_mult_closure
130	instantiation	157, 260	⊢
	:
131	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonneg
132	instantiation	158, 159, 206, 160	⊢
	: , :
133	theorem		⊢
	proveit.linear_algebra.inner_products.inner_prod_scalar_mult_right
134	instantiation	161, 260	⊢
	:
135	theorem		⊢
	proveit.physics.quantum.algebra.ket_zero_in_qubit_space
136	theorem		⊢
	proveit.physics.quantum.algebra.ket_one_in_qubit_space
137	instantiation	235, 162	⊢
	: , : , :
138	instantiation	163, 186, 164, 165*	,  ⊢
	: , :
139	instantiation	235, 166	,  ⊢
	: , : , :
140	instantiation	218, 167	⊢
	:
141	instantiation	253, 168, 169	⊢
	: , : , :
142	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
143	theorem		⊢
	proveit.numbers.division.mult_frac_cancel_denom_left
144	instantiation	326, 171, 170	⊢
	: , : , :
145	instantiation	326, 171, 184	⊢
	: , : , :
146	instantiation	235, 172	⊢
	: , : , :
147	instantiation	235, 173	⊢
	: , : , :
148	instantiation	220, 200	⊢
	:
149	theorem		⊢
	proveit.numbers.addition.add_nat_pos_from_nonneg
150	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
151	instantiation	174, 278, 311, 325, 279, 175, 201, 200, 203	⊢
	: , : , : , : , : , :
152	instantiation	176, 203, 200, 204	⊢
	: , : , :
153	instantiation	177, 203	⊢
	:
154	instantiation	178, 222, 221, 223, 179, 180*, 181*	⊢
	: , : , :
155	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_12
156	theorem		⊢
	proveit.numbers.addition.elim_zero_left
157	theorem		⊢
	proveit.linear_algebra.complex_vec_set_is_vec_space
158	theorem		⊢
	proveit.numbers.division.div_real_pos_closure
159	instantiation	326, 246, 182	⊢
	: , : , :
160	instantiation	183, 184	⊢
	:
161	theorem		⊢
	proveit.linear_algebra.inner_products.complex_vec_set_is_inner_prod_space
162	theorem		⊢
	proveit.physics.quantum.algebra.ket_zero_and_one_have_zero_inner_prod
163	axiom		⊢
	proveit.linear_algebra.scalar_multiplication.scalar_mult_extends_number_mult
164	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.zero_is_complex
165	instantiation	185, 186	,  ⊢
	:
166	instantiation	187, 188, 189*	,  ⊢
	:
167	instantiation	190, 200, 236	⊢
	: , : , :
168	instantiation	235, 191	⊢
	: , : , :
169	instantiation	235, 192	⊢
	: , : , :
170	instantiation	326, 193, 194	⊢
	: , : , :
171	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
172	instantiation	235, 195	⊢
	: , : , :
173	instantiation	253, 196, 197	⊢
	: , : , :
174	theorem		⊢
	proveit.numbers.addition.disassociation
175	instantiation	290	⊢
	: , :
176	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_31
177	theorem		⊢
	proveit.numbers.negation.double_negation
178	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
179	instantiation	198, 328	⊢
	:
180	instantiation	199, 200, 201	⊢
	: , :
181	instantiation	202, 203, 204	⊢
	: , :
182	instantiation	326, 259, 239	⊢
	: , : , :
183	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
184	instantiation	326, 205, 206	⊢
	: , : , :
185	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
186	instantiation	296, 207, 208	,  ⊢
	: , :
187	theorem		⊢
	proveit.numbers.absolute_value.complex_unit_length
188	instantiation	261, 209, 210	,  ⊢
	: , : , :
189	instantiation	211, 212	,  ⊢
	: , :
190	theorem		⊢
	proveit.logic.equality.sub_left_side_into
191	instantiation	253, 213, 215	⊢
	: , : , :
192	instantiation	253, 214, 215	⊢
	: , : , :
193	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
194	instantiation	326, 216, 217	⊢
	: , : , :
195	instantiation	218, 241	⊢
	:
196	instantiation	235, 219	⊢
	: , : , :
197	instantiation	220, 241	⊢
	:
198	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_lower_bound
199	theorem		⊢
	proveit.numbers.addition.commutation
200	instantiation	326, 306, 221	⊢
	: , : , :
201	instantiation	326, 306, 222	⊢
	: , : , :
202	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_basic
203	instantiation	326, 306, 223	⊢
	: , : , :
204	instantiation	224	⊢
	:
205	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
206	instantiation	225, 226	⊢
	:
207	instantiation	326, 306, 227	⊢
	: , : , :
208	instantiation	261, 228, 229	,  ⊢
	: , : , :
209	instantiation	292, 282, 230	,  ⊢
	: , :
210	instantiation	253, 231, 232	,  ⊢
	: , : , :
211	theorem		⊢
	proveit.logic.equality.equals_reversal
212	instantiation	235, 233	,  ⊢
	: , : , :
213	instantiation	235, 234	⊢
	: , : , :
214	instantiation	235, 236	⊢
	: , : , :
215	instantiation	237, 297	⊢
	:
216	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
217	instantiation	326, 238, 239	⊢
	: , : , :
218	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
219	instantiation	240, 241	⊢
	:
220	theorem		⊢
	proveit.numbers.division.frac_one_denom
221	instantiation	326, 309, 242	⊢
	: , : , :
222	instantiation	326, 309, 243	⊢
	: , : , :
223	instantiation	244, 245, 328	⊢
	: , : , :
224	axiom		⊢
	proveit.logic.equality.equals_reflexivity
225	theorem		⊢
	proveit.numbers.exponentiation.sqrt_real_pos_closure
226	instantiation	326, 246, 247	⊢
	: , : , :
227	instantiation	326, 299, 248	⊢
	: , : , :
228	instantiation	287, 264, 249	,  ⊢
	: , :
229	instantiation	253, 250, 251	,  ⊢
	: , : , :
230	instantiation	292, 252, 291	,  ⊢
	: , :
231	instantiation	277, 325, 311, 278, 271, 279, 264, 289, 281	,  ⊢
	: , : , : , : , : , :
232	instantiation	277, 278, 311, 279, 270, 271, 297, 275, 289, 281	,  ⊢
	: , : , : , : , : , :
233	instantiation	253, 254, 255	,  ⊢
	: , : , :
234	theorem		⊢
	proveit.physics.quantum.algebra.ket_zero_norm
235	axiom		⊢
	proveit.logic.equality.substitution
236	theorem		⊢
	proveit.physics.quantum.algebra.ket_one_norm
237	theorem		⊢
	proveit.numbers.exponentiation.exponentiated_one
238	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
239	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
240	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
241	instantiation	256, 297	⊢
	:
242	instantiation	326, 312, 321	⊢
	: , : , :
243	instantiation	326, 312, 320	⊢
	: , : , :
244	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
245	instantiation	257, 258	⊢
	: , :
246	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
247	instantiation	326, 259, 260	⊢
	: , : , :
248	theorem		⊢
	proveit.numbers.number_sets.real_numbers.e_is_real_pos
249	instantiation	261, 262, 263	,  ⊢
	: , : , :
250	instantiation	277, 325, 265, 278, 266, 279, 264, 288, 289, 281	,  ⊢
	: , : , : , : , : , :
251	instantiation	277, 278, 311, 265, 279, 270, 266, 297, 275, 288, 289, 281	,  ⊢
	: , : , : , : , : , :
252	instantiation	267, 302, 268	,  ⊢
	: , :
253	axiom		⊢
	proveit.logic.equality.equals_transitivity
254	instantiation	269, 278, 311, 279, 270, 271, 297, 275, 288, 289, 281	,  ⊢
	: , : , : , : , : , : , :
255	instantiation	272, 325, 273, 278, 274, 279, 288, 297, 275, 289, 281	,  ⊢
	: , : , : , : , : , :
256	theorem		⊢
	proveit.numbers.exponentiation.sqrt_complex_closure
257	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
258	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
259	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
260	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
261	theorem		⊢
	proveit.logic.equality.sub_right_side_into
262	instantiation	287, 276, 281	,  ⊢
	: , :
263	instantiation	277, 278, 311, 325, 279, 280, 288, 289, 281	,  ⊢
	: , : , : , : , : , :
264	instantiation	326, 306, 282	⊢
	: , : , :
265	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
266	instantiation	283	⊢
	: , : , :
267	theorem		⊢
	proveit.numbers.exponentiation.exp_real_closure_nat_power
268	instantiation	284, 285	,  ⊢
	:
269	theorem		⊢
	proveit.numbers.multiplication.leftward_commutation
270	instantiation	290	⊢
	: , :
271	instantiation	290	⊢
	: , :
272	theorem		⊢
	proveit.numbers.multiplication.association
273	theorem		⊢
	proveit.numbers.numerals.decimals.nat4
274	instantiation	286	⊢
	: , : , : , :
275	instantiation	326, 306, 293	⊢
	: , : , :
276	instantiation	287, 288, 289	,  ⊢
	: , :
277	theorem		⊢
	proveit.numbers.multiplication.disassociation
278	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
279	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
280	instantiation	290	⊢
	: , :
281	instantiation	326, 306, 291	⊢
	: , : , :
282	instantiation	292, 302, 293	⊢
	: , :
283	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
284	theorem		⊢
	proveit.numbers.negation.nat_closure
285	instantiation	294, 313, 295	,  ⊢
	:
286	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_4_typical_eq
287	theorem		⊢
	proveit.numbers.multiplication.mult_complex_closure_bin
288	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.i_is_complex
289	instantiation	296, 297, 298	,  ⊢
	: , :
290	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
291	theorem		⊢
	proveit.physics.quantum.QPE._phase_is_real
292	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
293	instantiation	326, 299, 300	⊢
	: , : , :
294	theorem		⊢
	proveit.numbers.number_sets.integers.nonpos_int_is_int_nonpos
295	instantiation	301, 317, 318, 315	,  ⊢
	: , : , :
296	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
297	instantiation	326, 306, 302	⊢
	: , : , :
298	instantiation	303, 304	,  ⊢
	:
299	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
300	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pi_is_real_pos
301	theorem		⊢
	proveit.numbers.number_sets.integers.interval_upper_bound
302	instantiation	326, 309, 305	⊢
	: , : , :
303	theorem		⊢
	proveit.numbers.negation.complex_closure
304	instantiation	326, 306, 307	,  ⊢
	: , : , :
305	instantiation	326, 312, 308	⊢
	: , : , :
306	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
307	instantiation	326, 309, 310	,  ⊢
	: , : , :
308	instantiation	326, 324, 311	⊢
	: , : , :
309	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
310	instantiation	326, 312, 313	,  ⊢
	: , : , :
311	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
312	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
313	instantiation	326, 314, 315	,  ⊢
	: , : , :
314	instantiation	316, 317, 318	⊢
	: , :
315	assumption		⊢
316	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
317	instantiation	319, 320, 321	⊢
	: , :
318	theorem		⊢
	proveit.numbers.number_sets.integers.zero_is_int
319	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
320	instantiation	322, 323	⊢
	:
321	instantiation	326, 324, 325	⊢
	: , : , :
322	theorem		⊢
	proveit.numbers.negation.int_closure
323	instantiation	326, 327, 328	⊢
	: , : , :
324	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
325	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
326	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
327	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
328	assumption		⊢
*equality replacement requirements