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