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	⊢
	: , : , :
1	reference	138	⊢
2	instantiation	138, 4, 5, 6*	⊢
	: , : , :
3	modus ponens	7, 56	⊢
4	instantiation	138, 8, 9	⊢
	: , : , :
5	instantiation	10, 228	⊢
	:
6	instantiation	162, 11, 12	⊢
	: , : , :
7	instantiation	13, 218, 226, 153, 85, 154, 14*	⊢
	: , : , : , : , : , : , : , :
8	instantiation	15, 123	⊢
	:
9	axiom		⊢
	proveit.physics.quantum.QPE._Psi_def
10	theorem		⊢
	proveit.physics.quantum.QPE._psi_t_formula
11	instantiation	171, 16	⊢
	: , : , :
12	instantiation	162, 17, 18	⊢
	: , : , :
13	theorem		⊢
	proveit.physics.quantum.algebra.qmult_distribution_over_summation
14	instantiation	162, 19, 20	⊢
	: , : , :
15	axiom		⊢
	proveit.physics.quantum.QPE._alpha_m_def
16	instantiation	50, 21, 22, 23	⊢
	: , : , : , :
17	instantiation	162, 24, 25	⊢
	: , : , :
18	instantiation	61, 62, 95, 26, 56, 27*	⊢
	: , : , :
19	instantiation	171, 28, 29*, 30*	⊢
	: , : , :
20	modus ponens	31, 32	⊢
21	instantiation	59, 95, 231, 153, 154, 33	⊢
	: , : , : , : , : , :
22	instantiation	60, 231, 95, 33	⊢
	: , : , : , : , :
23	instantiation	61, 220, 95, 55, 33	⊢
	: , : , :
24	instantiation	162, 34, 35	⊢
	: , : , :
25	instantiation	60, 231, 226, 95, 55, 56	⊢
	: , : , : , : , :
26	instantiation	158	⊢
	: , : , :
27	instantiation	70, 95, 36	⊢
	: , :
28	modus ponens	37, 38	⊢
29	instantiation	39, 194	⊢
	: , :
30	instantiation	39, 194	⊢
	: , :
31	instantiation	40, 218	⊢
	: , : , : , :
32	generalization	41	⊢
33	instantiation	75, 107, 98, 72	⊢
	: , : , : , :
34	instantiation	59, 95, 231, 153, 154, 42	⊢
	: , : , : , : , : , :
35	instantiation	54, 226, 220, 153, 81, 55, 154, 43	⊢
	: , : , : , : , : , :
36	instantiation	118, 44, 67	⊢
	: , : , :
37	instantiation	45, 218	⊢
	: , : , : , : , : , : , :
38	generalization	46	⊢
39	theorem		⊢
	proveit.core_expr_types.conditionals.satisfied_condition_reduction
40	axiom		⊢
	proveit.linear_algebra.addition.scalar_sum_extends_number_sum
41	instantiation	165, 92, 71	,  ⊢
	: , :
42	instantiation	118, 56, 47	⊢
	: , : , :
43	instantiation	118, 48, 49	⊢
	: , : , :
44	instantiation	82, 107, 108, 83, 72	⊢
	: , : , : , :
45	theorem		⊢
	proveit.core_expr_types.lambda_maps.general_lambda_substitution
46	instantiation	50, 51, 52, 53	,  ⊢
	: , : , : , :
47	instantiation	54, 231, 220, 153, 55, 154, 56	⊢
	: , : , : , : , : , :
48	instantiation	75, 107, 108, 57, 72	⊢
	: , : , : , :
49	instantiation	84, 62, 58	⊢
	: , : , :
50	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
51	instantiation	59, 92, 226, 153, 85, 154, 64	,  ⊢
	: , : , : , : , : , :
52	instantiation	60, 226, 231, 92, 85, 64	,  ⊢
	: , : , : , : , :
53	instantiation	61, 62, 92, 63, 64, 65*	,  ⊢
	: , : , :
54	theorem		⊢
	proveit.physics.quantum.algebra.qmult_disassociation
55	instantiation	168	⊢
	: , :
56	instantiation	118, 66, 67	⊢
	: , : , :
57	instantiation	106, 107, 108, 68	⊢
	: , : , :
58	instantiation	158	⊢
	: , : , :
59	theorem		⊢
	proveit.physics.quantum.algebra.scalar_mult_absorption
60	theorem		⊢
	proveit.physics.quantum.algebra.qmult_pulling_scalar_out_front
61	theorem		⊢
	proveit.physics.quantum.algebra.scalar_mult_factorization
62	theorem		⊢
	proveit.numbers.numerals.decimals.posnat3
63	instantiation	158	⊢
	: , : , :
64	instantiation	118, 69, 77	,  ⊢
	: , : , :
65	instantiation	70, 92, 71	,  ⊢
	: , :
66	instantiation	75, 107, 108, 83, 72	⊢
	: , : , : , :
67	instantiation	84, 220, 85	⊢
	: , : , :
68	instantiation	118, 73, 74	⊢
	: , : , :
69	instantiation	75, 107, 108, 83, 93	,  ⊢
	: , : , : , :
70	axiom		⊢
	proveit.linear_algebra.scalar_multiplication.scalar_mult_extends_number_mult
71	instantiation	118, 76, 77	,  ⊢
	: , : , :
72	modus ponens	78, 79	⊢
73	instantiation	96, 107, 108, 80, 98	⊢
	: , : , : , : , :
74	instantiation	84, 220, 81	⊢
	: , : , :
75	theorem		⊢
	proveit.physics.quantum.algebra.qmult_op_ket_in_QmultCodomain
76	instantiation	82, 107, 108, 83, 93	,  ⊢
	: , : , : , :
77	instantiation	84, 220, 85	⊢
	: , : , :
78	instantiation	86, 218, 91	⊢
	: , : , : , : , : , :
79	generalization	87	⊢
80	instantiation	106, 107, 108, 88	⊢
	: , : , :
81	instantiation	168	⊢
	: , :
82	theorem		⊢
	proveit.physics.quantum.algebra.qmult_op_ket_is_ket
83	instantiation	106, 107, 108, 89	⊢
	: , : , :
84	axiom		⊢
	proveit.physics.quantum.algebra.multi_qmult_def
85	instantiation	168	⊢
	: , :
86	theorem		⊢
	proveit.linear_algebra.addition.summation_closure
87	instantiation	90, 91, 92, 93	⊢
	: , : , : , :
88	instantiation	94, 95, 107, 108, 97	⊢
	: , : , : , :
89	instantiation	96, 107, 108, 97, 98	⊢
	: , : , : , : , :
90	theorem		⊢
	proveit.linear_algebra.scalar_multiplication.scalar_mult_closure
91	instantiation	99, 136	⊢
	:
92	instantiation	116, 100, 101	⊢
	: , :
93	instantiation	102, 228, 194	⊢
	: , :
94	theorem		⊢
	proveit.physics.quantum.algebra.qmult_op_complex_closure
95	instantiation	131, 103, 104, 105	⊢
	: , :
96	theorem		⊢
	proveit.physics.quantum.algebra.qmult_op_op_is_op
97	instantiation	106, 107, 108, 109	⊢
	: , : , :
98	instantiation	110, 136, 111	⊢
	: , : , :
99	theorem		⊢
	proveit.linear_algebra.complex_vec_set_is_vec_space
100	instantiation	229, 197, 112	⊢
	: , : , :
101	instantiation	138, 113, 114	⊢
	: , : , :
102	theorem		⊢
	proveit.physics.quantum.algebra.num_ket_in_register_space
103	instantiation	229, 197, 115	⊢
	: , : , :
104	instantiation	116, 189, 117	⊢
	: , :
105	instantiation	118, 119, 120	⊢
	: , : , :
106	theorem		⊢
	proveit.physics.quantum.algebra.qmult_op_is_linmap
107	instantiation	121, 136	⊢
	:
108	theorem		⊢
	proveit.linear_algebra.inner_products.complex_set_is_hilbert_space
109	instantiation	122, 228, 123	⊢
	: , :
110	theorem		⊢
	proveit.physics.quantum.algebra.qmult_matrix_is_linmap
111	instantiation	190, 124, 125	⊢
	: , : , :
112	instantiation	229, 178, 126	⊢
	: , : , :
113	instantiation	165, 141, 127	⊢
	: , :
114	instantiation	162, 128, 129	⊢
	: , : , :
115	instantiation	229, 208, 130	⊢
	: , : , :
116	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
117	instantiation	131, 174, 189, 148	⊢
	: , :
118	theorem		⊢
	proveit.logic.equality.sub_left_side_into
119	instantiation	132, 196, 133	⊢
	: , :
120	instantiation	171, 134	⊢
	: , : , :
121	theorem		⊢
	proveit.linear_algebra.inner_products.complex_vec_set_is_hilbert_space
122	theorem		⊢
	proveit.physics.quantum.algebra.num_bra_is_lin_map
123	assumption		⊢
124	instantiation	135, 136	⊢
	:
125	instantiation	137, 228	⊢
	:
126	theorem		⊢
	proveit.numbers.number_sets.real_numbers.e_is_real_pos
127	instantiation	138, 139, 140	⊢
	: , : , :
128	instantiation	152, 231, 142, 153, 144, 154, 141, 166, 167, 156	⊢
	: , : , : , : , : , :
129	instantiation	152, 153, 226, 142, 154, 143, 144, 189, 157, 166, 167, 156	⊢
	: , : , : , : , : , :
130	instantiation	229, 217, 225	⊢
	: , : , :
131	theorem		⊢
	proveit.numbers.division.div_complex_closure
132	theorem		⊢
	proveit.numbers.exponentiation.exp_rational_non_zero__not_zero
133	instantiation	229, 145, 146	⊢
	: , : , :
134	instantiation	147, 174, 189, 148, 149*	⊢
	: , :
135	theorem		⊢
	proveit.linear_algebra.matrices.unitaries_are_matrices
136	instantiation	150, 226, 223	⊢
	: , :
137	theorem		⊢
	proveit.physics.quantum.QFT.invFT_is_unitary
138	theorem		⊢
	proveit.logic.equality.sub_right_side_into
139	instantiation	165, 151, 156	⊢
	: , :
140	instantiation	152, 153, 226, 231, 154, 155, 166, 167, 156	⊢
	: , : , : , : , : , :
141	instantiation	165, 189, 157	⊢
	: , :
142	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
143	instantiation	168	⊢
	: , :
144	instantiation	158	⊢
	: , : , :
145	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational_nonzero
146	instantiation	159, 202, 160	⊢
	: , :
147	theorem		⊢
	proveit.numbers.division.div_as_mult
148	instantiation	161, 220	⊢
	:
149	instantiation	162, 163, 164	⊢
	: , : , :
150	theorem		⊢
	proveit.numbers.exponentiation.exp_natpos_closure
151	instantiation	165, 166, 167	⊢
	: , :
152	theorem		⊢
	proveit.numbers.multiplication.disassociation
153	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
154	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
155	instantiation	168	⊢
	: , :
156	instantiation	229, 197, 169	⊢
	: , : , :
157	instantiation	229, 197, 170	⊢
	: , : , :
158	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
159	theorem		⊢
	proveit.numbers.multiplication.mult_rational_pos_closure_bin
160	instantiation	229, 219, 228	⊢
	: , : , :
161	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
162	axiom		⊢
	proveit.logic.equality.equals_transitivity
163	instantiation	171, 172	⊢
	: , : , :
164	instantiation	173, 174, 175	⊢
	: , :
165	theorem		⊢
	proveit.numbers.multiplication.mult_complex_closure_bin
166	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.i_is_complex
167	instantiation	229, 197, 176	⊢
	: , : , :
168	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
169	instantiation	229, 208, 177	⊢
	: , : , :
170	instantiation	229, 178, 179	⊢
	: , : , :
171	axiom		⊢
	proveit.logic.equality.substitution
172	instantiation	180, 181, 218, 182*	⊢
	: , :
173	theorem		⊢
	proveit.numbers.multiplication.commutation
174	instantiation	229, 197, 183	⊢
	: , : , :
175	instantiation	229, 197, 184	⊢
	: , : , :
176	theorem		⊢
	proveit.physics.quantum.QPE._phase_is_real
177	instantiation	229, 217, 185	⊢
	: , : , :
178	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
179	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pi_is_real_pos
180	theorem		⊢
	proveit.numbers.exponentiation.neg_power_as_div
181	instantiation	229, 186, 187	⊢
	: , : , :
182	instantiation	188, 189	⊢
	:
183	instantiation	190, 191, 228	⊢
	: , : , :
184	instantiation	229, 208, 192	⊢
	: , : , :
185	instantiation	229, 193, 194	⊢
	: , : , :
186	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
187	instantiation	229, 195, 196	⊢
	: , : , :
188	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
189	instantiation	229, 197, 198	⊢
	: , : , :
190	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
191	instantiation	199, 200	⊢
	: , :
192	instantiation	229, 201, 202	⊢
	: , : , :
193	instantiation	203, 204, 205	⊢
	: , :
194	assumption		⊢
195	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
196	instantiation	229, 206, 207	⊢
	: , : , :
197	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
198	instantiation	229, 208, 209	⊢
	: , : , :
199	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
200	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
201	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
202	instantiation	210, 211, 212	⊢
	: , :
203	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
204	theorem		⊢
	proveit.numbers.number_sets.integers.zero_is_int
205	instantiation	213, 214, 215	⊢
	: , :
206	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
207	instantiation	229, 216, 220	⊢
	: , : , :
208	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
209	instantiation	229, 217, 222	⊢
	: , : , :
210	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
211	instantiation	229, 219, 218	⊢
	: , : , :
212	instantiation	229, 219, 220	⊢
	: , : , :
213	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
214	instantiation	221, 222, 223	⊢
	: , :
215	instantiation	224, 225	⊢
	:
216	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
217	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
218	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
219	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
220	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
221	theorem		⊢
	proveit.numbers.exponentiation.exp_int_closure
222	instantiation	229, 230, 226	⊢
	: , : , :
223	instantiation	229, 227, 228	⊢
	: , : , :
224	theorem		⊢
	proveit.numbers.negation.int_closure
225	instantiation	229, 230, 231	⊢
	: , : , :
226	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
227	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
228	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
229	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
230	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
231	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements