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