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