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, 13*	⊢
	: , : , : , : , :
1	theorem		⊢
	proveit.physics.quantum.circuits.phase_kickbacks_on_register
2	reference	110	⊢
3	reference	191	⊢
4	instantiation	93, 14, 165, 18	⊢
	: , : , : , :
5	modus ponens	15, 16	⊢
6	axiom		⊢
	proveit.physics.quantum.QPE._u_ket_register
7	axiom		⊢
	proveit.physics.quantum.QPE._normalized_ket_u
8	instantiation	93, 17, 165, 18	⊢
	: , : , : , :
9	modus ponens	19, 20	⊢
10	reference	180	⊢
11	reference	188	⊢
12	instantiation	21, 182, 22, 23, 24, 25*	⊢
	: , : , : , : , : , :
13	instantiation	26, 115, 147, 139, 27, 28, 141, 50, 29, 30, 31, 32	,  ⊢
	: , : , : , : , : , :
14	instantiation	35, 33, 37	⊢
	: , : , :
15	instantiation	39, 167, 168, 40	⊢
	: , : , : , :
16	generalization	34	⊢
17	instantiation	35, 36, 37	⊢
	: , : , :
18	instantiation	126, 38	⊢
	: , :
19	instantiation	39, 167, 168, 40	⊢
	: , : , : , :
20	generalization	41	⊢
21	theorem		⊢
	proveit.core_expr_types.tuples.shift_equivalence
22	instantiation	42, 43, 83	⊢
	: , :
23	instantiation	126, 44	⊢
	: , :
24	instantiation	126, 45	⊢
	: , :
25	instantiation	123, 46, 47	,  ⊢
	: , : , :
26	theorem		⊢
	proveit.numbers.multiplication.disassociation
27	instantiation	136	⊢
	: , : , :
28	instantiation	153	⊢
	: , :
29	instantiation	189, 171, 48	⊢
	: , : , :
30	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.i_is_complex
31	instantiation	49, 50, 51	,  ⊢
	: , :
32	instantiation	189, 171, 67	⊢
	: , : , :
33	instantiation	55, 56	⊢
	: , : , :
34	instantiation	52, 66, 53, 54	,  ⊢
	: , : , :
35	theorem		⊢
	proveit.logic.equality.sub_right_side_into
36	instantiation	55, 56	⊢
	: , : , :
37	instantiation	123, 57, 58	⊢
	: , : , :
38	instantiation	59, 60	⊢
	: , :
39	theorem		⊢
	proveit.logic.booleans.conjunction.conjunction_from_quantification
40	instantiation	149, 61, 99, 172, 62, 63*, 64*	⊢
	: , : , :
41	instantiation	65, 66, 67, 68	,  ⊢
	: , : , : , :
42	theorem		⊢
	proveit.numbers.addition.add_nat_closure_bin
43	instantiation	189, 109, 69	⊢
	: , : , :
44	instantiation	123, 70, 71	⊢
	: , : , :
45	instantiation	123, 72, 73	⊢
	: , : , :
46	instantiation	138, 139, 147, 186, 141, 74, 75, 162, 164	,  ⊢
	: , : , : , : , : , :
47	instantiation	118, 164, 75, 165	,  ⊢
	: , : , :
48	instantiation	189, 76, 77	⊢
	: , : , :
49	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
50	instantiation	189, 171, 78	⊢
	: , : , :
51	instantiation	137, 79	,  ⊢
	:
52	theorem		⊢
	proveit.linear_algebra.matrices.exponentiation.U_closure
53	instantiation	96, 147, 80	⊢
	: , :
54	axiom		⊢
	proveit.physics.quantum.QPE._unitary_U
55	theorem		⊢
	proveit.core_expr_types.tuples.range_len
56	instantiation	81, 115, 82, 139, 83, 186	⊢
	: , :
57	instantiation	105, 84	⊢
	: , : , :
58	instantiation	93, 85, 86, 87	⊢
	: , : , : , :
59	theorem		⊢
	proveit.core_expr_types.tuples.range_from1_len
60	instantiation	189, 109, 191	⊢
	: , : , :
61	instantiation	189, 176, 88	⊢
	: , : , :
62	instantiation	89, 90	⊢
	: , :
63	instantiation	123, 91, 92	⊢
	: , : , :
64	instantiation	93, 94, 119, 95	⊢
	: , : , : , :
65	theorem		⊢
	proveit.linear_algebra.matrices.exponentiation.unital2pi_eigen_exp_application
66	instantiation	96, 147, 97	,  ⊢
	: , :
67	instantiation	98, 99, 169, 100	⊢
	: , : , :
68	axiom		⊢
	proveit.physics.quantum.QPE._eigen_uu
69	instantiation	101, 186, 139, 141, 102	⊢
	: , : , : , : , :
70	instantiation	105, 113	⊢
	: , : , :
71	instantiation	123, 103, 104	⊢
	: , : , :
72	instantiation	105, 113	⊢
	: , : , :
73	instantiation	143, 164	⊢
	:
74	instantiation	153	⊢
	: , :
75	instantiation	189, 171, 106	,  ⊢
	: , : , :
76	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
77	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pi_is_real_pos
78	instantiation	189, 176, 107	⊢
	: , : , :
79	instantiation	189, 171, 108	,  ⊢
	: , : , :
80	instantiation	189, 109, 110	⊢
	: , : , :
81	theorem		⊢
	proveit.numbers.addition.add_nat_closure
82	instantiation	136	⊢
	: , : , :
83	instantiation	128, 111	⊢
	:
84	instantiation	112, 162, 161, 113*	⊢
	: , :
85	instantiation	138, 186, 147, 114, 121, 164, 117, 161	⊢
	: , : , : , : , : , :
86	instantiation	122, 139, 115, 141, 116, 164, 117, 161	⊢
	: , : , : , :
87	instantiation	118, 161, 164, 119	⊢
	: , : , :
88	instantiation	189, 181, 167	⊢
	: , : , :
89	theorem		⊢
	proveit.numbers.ordering.relax_less
90	instantiation	120, 191	⊢
	:
91	instantiation	138, 186, 147, 139, 140, 141, 121, 162, 161	⊢
	: , : , : , : , : , :
92	instantiation	122, 139, 147, 141, 140, 162, 161	⊢
	: , : , : , :
93	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
94	instantiation	123, 124, 125	⊢
	: , : , :
95	instantiation	126, 127	⊢
	: , :
96	theorem		⊢
	proveit.numbers.exponentiation.exp_natpos_closure
97	instantiation	128, 129	,  ⊢
	:
98	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_co__is__real
99	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
100	axiom		⊢
	proveit.physics.quantum.QPE._phase_in_interval
101	theorem		⊢
	proveit.numbers.addition.add_nat_pos_from_nonneg
102	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
103	instantiation	138, 139, 147, 186, 141, 140, 162, 161, 164	⊢
	: , : , : , : , : , :
104	instantiation	130, 164, 161, 165	⊢
	: , : , :
105	axiom		⊢
	proveit.logic.equality.substitution
106	instantiation	189, 176, 131	,  ⊢
	: , : , :
107	instantiation	189, 181, 132	⊢
	: , : , :
108	instantiation	189, 176, 133	,  ⊢
	: , : , :
109	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
110	axiom		⊢
	proveit.physics.quantum.QPE._s_in_nat_pos
111	instantiation	144, 167, 134	⊢
	:
112	theorem		⊢
	proveit.numbers.negation.distribute_neg_through_binary_sum
113	instantiation	135, 164	⊢
	:
114	instantiation	153	⊢
	: , :
115	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
116	instantiation	136	⊢
	: , : , :
117	instantiation	137, 161	⊢
	:
118	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
119	instantiation	173	⊢
	:
120	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
121	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.zero_is_complex
122	theorem		⊢
	proveit.numbers.addition.elim_zero_any
123	axiom		⊢
	proveit.logic.equality.equals_transitivity
124	instantiation	138, 186, 147, 139, 140, 141, 164, 162, 161	⊢
	: , : , : , : , : , :
125	instantiation	142, 164, 161, 165	⊢
	: , : , :
126	theorem		⊢
	proveit.logic.equality.equals_reversal
127	instantiation	143, 161	⊢
	:
128	theorem		⊢
	proveit.numbers.negation.nat_closure
129	instantiation	144, 148, 145	,  ⊢
	:
130	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_31
131	instantiation	189, 181, 146	,  ⊢
	: , : , :
132	instantiation	189, 185, 147	⊢
	: , : , :
133	instantiation	189, 181, 148	,  ⊢
	: , : , :
134	instantiation	149, 170, 169, 172, 150, 151*, 152*	⊢
	: , : , :
135	theorem		⊢
	proveit.numbers.negation.double_negation
136	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
137	theorem		⊢
	proveit.numbers.negation.complex_closure
138	theorem		⊢
	proveit.numbers.addition.disassociation
139	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
140	instantiation	153	⊢
	: , :
141	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
142	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_12
143	theorem		⊢
	proveit.numbers.addition.elim_zero_left
144	theorem		⊢
	proveit.numbers.number_sets.integers.nonpos_int_is_int_nonpos
145	instantiation	154, 167, 168, 158	,  ⊢
	: , : , :
146	instantiation	189, 155, 156	,  ⊢
	: , : , :
147	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
148	instantiation	189, 157, 158	,  ⊢
	: , : , :
149	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
150	instantiation	159, 191	⊢
	:
151	instantiation	160, 161, 162	⊢
	: , :
152	instantiation	163, 164, 165	⊢
	: , :
153	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
154	theorem		⊢
	proveit.numbers.number_sets.integers.interval_upper_bound
155	instantiation	166, 180, 188	⊢
	: , :
156	assumption		⊢
157	instantiation	166, 167, 168	⊢
	: , :
158	assumption		⊢
159	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_lower_bound
160	theorem		⊢
	proveit.numbers.addition.commutation
161	instantiation	189, 171, 169	⊢
	: , : , :
162	instantiation	189, 171, 170	⊢
	: , : , :
163	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_basic
164	instantiation	189, 171, 172	⊢
	: , : , :
165	instantiation	173	⊢
	:
166	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
167	instantiation	174, 182, 180	⊢
	: , :
168	theorem		⊢
	proveit.numbers.number_sets.integers.zero_is_int
169	instantiation	189, 176, 175	⊢
	: , : , :
170	instantiation	189, 176, 177	⊢
	: , : , :
171	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
172	instantiation	178, 179, 191	⊢
	: , : , :
173	axiom		⊢
	proveit.logic.equality.equals_reflexivity
174	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
175	instantiation	189, 181, 180	⊢
	: , : , :
176	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
177	instantiation	189, 181, 182	⊢
	: , : , :
178	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
179	instantiation	183, 184	⊢
	: , :
180	instantiation	189, 185, 186	⊢
	: , : , :
181	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
182	instantiation	187, 188	⊢
	:
183	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
184	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
185	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
186	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
187	theorem		⊢
	proveit.numbers.negation.int_closure
188	instantiation	189, 190, 191	⊢
	: , : , :
189	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
190	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
191	assumption		⊢
*equality replacement requirements