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