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	theorem		⊢
	proveit.physics.quantum.circuits.rhs_prob_via_equiv
2	instantiation	4, 108, 199, 5, 6, 7, 8, 9, 10, 188, 196, 11, 12*	⊢
	: , : , : , : , :
3	modus ponens	13, 14	⊢
4	theorem		⊢
	proveit.physics.quantum.circuits.phase_kickbacks_on_register
5	instantiation	125, 15, 177, 54	⊢
	: , : , : , :
6	modus ponens	16, 17	⊢
7	axiom		⊢
	proveit.physics.quantum.QPE._u_ket_register
8	axiom		⊢
	proveit.physics.quantum.QPE._normalized_ket_u
9	instantiation	125, 18, 177, 54	⊢
	: , : , : , :
10	modus ponens	19, 20	⊢
11	instantiation	21, 190, 22, 23, 24, 25*	⊢
	: , : , : , : , : , :
12	instantiation	26, 146, 141, 145, 27, 28, 147, 51, 29, 30, 31, 32	,  ⊢
	: , : , : , : , : , :
13	instantiation	33, 194, 107, 34	⊢
	: , : , : , : , : , : , : , :
14	instantiation	35, 36	⊢
	: , :
15	instantiation	78, 37, 80	⊢
	: , : , :
16	instantiation	40, 164, 165, 41	⊢
	: , : , : , :
17	generalization	38	⊢
18	instantiation	78, 39, 80	⊢
	: , : , :
19	instantiation	40, 164, 165, 41	⊢
	: , : , : , :
20	generalization	42	⊢
21	theorem		⊢
	proveit.core_expr_types.tuples.shift_equivalence
22	instantiation	43, 44, 122	⊢
	: , :
23	instantiation	113, 45	⊢
	: , :
24	instantiation	113, 46	⊢
	: , :
25	instantiation	110, 47, 48	,  ⊢
	: , : , :
26	theorem		⊢
	proveit.numbers.multiplication.disassociation
27	instantiation	161	⊢
	: , : , :
28	instantiation	160	⊢
	: , :
29	instantiation	197, 180, 49	⊢
	: , : , :
30	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.i_is_complex
31	instantiation	50, 51, 52	,  ⊢
	: , :
32	instantiation	197, 180, 65	⊢
	: , : , :
33	theorem		⊢
	proveit.physics.quantum.circuits.circuit_equiv_temporal_sub
34	instantiation	125, 53, 177, 54	⊢
	: , : , : , :
35	theorem		⊢
	proveit.physics.quantum.circuits.equiv_reversal
36	instantiation	55, 108, 199, 58	⊢
	: , : , :
37	instantiation	102, 103	⊢
	: , : , :
38	instantiation	56, 64, 57, 58	,  ⊢
	: , : , :
39	instantiation	102, 103	⊢
	: , : , :
40	theorem		⊢
	proveit.logic.booleans.conjunction.conjunction_from_quantification
41	instantiation	166, 59, 93, 181, 60, 61*, 62*	⊢
	: , : , :
42	instantiation	63, 64, 65, 66	,  ⊢
	: , : , : , :
43	theorem		⊢
	proveit.numbers.addition.add_nat_closure_bin
44	instantiation	197, 129, 67	⊢
	: , : , :
45	instantiation	110, 68, 69	⊢
	: , : , :
46	instantiation	110, 70, 71	⊢
	: , : , :
47	instantiation	140, 145, 141, 194, 147, 72, 73, 174, 176	,  ⊢
	: , : , : , : , : , :
48	instantiation	150, 176, 73, 177	,  ⊢
	: , : , :
49	instantiation	197, 74, 75	⊢
	: , : , :
50	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
51	instantiation	197, 180, 76	⊢
	: , : , :
52	instantiation	162, 77	,  ⊢
	:
53	instantiation	78, 79, 80	⊢
	: , : , :
54	instantiation	113, 81	⊢
	: , :
55	axiom		⊢
	proveit.physics.quantum.QPE.QPE1_def
56	theorem		⊢
	proveit.linear_algebra.matrices.exponentiation.U_closure
57	instantiation	90, 141, 82	⊢
	: , :
58	axiom		⊢
	proveit.physics.quantum.QPE._unitary_U
59	instantiation	197, 184, 83	⊢
	: , : , :
60	instantiation	84, 85	⊢
	: , :
61	instantiation	110, 86, 87	⊢
	: , : , :
62	instantiation	125, 88, 151, 89	⊢
	: , : , : , :
63	theorem		⊢
	proveit.linear_algebra.matrices.exponentiation.unital2pi_eigen_exp_application
64	instantiation	90, 141, 91	,  ⊢
	: , :
65	instantiation	92, 93, 178, 94	⊢
	: , : , :
66	axiom		⊢
	proveit.physics.quantum.QPE._eigen_uu
67	instantiation	95, 194, 145, 147, 96	⊢
	: , : , : , : , :
68	instantiation	123, 139	⊢
	: , : , :
69	instantiation	110, 97, 98	⊢
	: , : , :
70	instantiation	123, 139	⊢
	: , : , :
71	instantiation	132, 176	⊢
	:
72	instantiation	160	⊢
	: , :
73	instantiation	197, 180, 99	,  ⊢
	: , : , :
74	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
75	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pi_is_real_pos
76	instantiation	197, 184, 100	⊢
	: , : , :
77	instantiation	197, 180, 101	,  ⊢
	: , : , :
78	theorem		⊢
	proveit.logic.equality.sub_right_side_into
79	instantiation	102, 103	⊢
	: , : , :
80	instantiation	110, 104, 105	⊢
	: , : , :
81	instantiation	106, 107	⊢
	: , :
82	instantiation	197, 129, 108	⊢
	: , : , :
83	instantiation	197, 189, 164	⊢
	: , : , :
84	theorem		⊢
	proveit.numbers.ordering.relax_less
85	instantiation	109, 199	⊢
	:
86	instantiation	140, 194, 141, 145, 130, 147, 143, 174, 173	⊢
	: , : , : , : , : , :
87	instantiation	144, 145, 141, 147, 130, 174, 173	⊢
	: , : , : , :
88	instantiation	110, 111, 112	⊢
	: , : , :
89	instantiation	113, 114	⊢
	: , :
90	theorem		⊢
	proveit.numbers.exponentiation.exp_natpos_closure
91	instantiation	136, 115	,  ⊢
	:
92	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_co__is__real
93	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
94	axiom		⊢
	proveit.physics.quantum.QPE._phase_in_interval
95	theorem		⊢
	proveit.numbers.addition.add_nat_pos_from_nonneg
96	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
97	instantiation	140, 145, 141, 194, 147, 130, 174, 173, 176	⊢
	: , : , : , : , : , :
98	instantiation	116, 176, 173, 177	⊢
	: , : , :
99	instantiation	197, 184, 117	,  ⊢
	: , : , :
100	instantiation	197, 189, 118	⊢
	: , : , :
101	instantiation	197, 184, 119	,  ⊢
	: , : , :
102	theorem		⊢
	proveit.core_expr_types.tuples.range_len
103	instantiation	120, 146, 121, 145, 122, 194	⊢
	: , :
104	instantiation	123, 124	⊢
	: , : , :
105	instantiation	125, 126, 127, 128	⊢
	: , : , : , :
106	theorem		⊢
	proveit.core_expr_types.tuples.range_from1_len
107	instantiation	197, 129, 199	⊢
	: , : , :
108	axiom		⊢
	proveit.physics.quantum.QPE._s_in_nat_pos
109	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
110	axiom		⊢
	proveit.logic.equality.equals_transitivity
111	instantiation	140, 194, 141, 145, 130, 147, 176, 174, 173	⊢
	: , : , : , : , : , :
112	instantiation	131, 176, 173, 177	⊢
	: , : , :
113	theorem		⊢
	proveit.logic.equality.equals_reversal
114	instantiation	132, 173	⊢
	:
115	instantiation	157, 135, 133	,  ⊢
	:
116	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_31
117	instantiation	197, 189, 134	,  ⊢
	: , : , :
118	instantiation	197, 193, 141	⊢
	: , : , :
119	instantiation	197, 189, 135	,  ⊢
	: , : , :
120	theorem		⊢
	proveit.numbers.addition.add_nat_closure
121	instantiation	161	⊢
	: , : , :
122	instantiation	136, 137	⊢
	:
123	axiom		⊢
	proveit.logic.equality.substitution
124	instantiation	138, 174, 173, 139*	⊢
	: , :
125	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
126	instantiation	140, 194, 141, 142, 143, 176, 149, 173	⊢
	: , : , : , : , : , :
127	instantiation	144, 145, 146, 147, 148, 176, 149, 173	⊢
	: , : , : , :
128	instantiation	150, 173, 176, 151	⊢
	: , : , :
129	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
130	instantiation	160	⊢
	: , :
131	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_12
132	theorem		⊢
	proveit.numbers.addition.elim_zero_left
133	instantiation	152, 164, 165, 156	,  ⊢
	: , : , :
134	instantiation	197, 153, 154	,  ⊢
	: , : , :
135	instantiation	197, 155, 156	,  ⊢
	: , : , :
136	theorem		⊢
	proveit.numbers.negation.nat_closure
137	instantiation	157, 164, 158	⊢
	:
138	theorem		⊢
	proveit.numbers.negation.distribute_neg_through_binary_sum
139	instantiation	159, 176	⊢
	:
140	theorem		⊢
	proveit.numbers.addition.disassociation
141	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
142	instantiation	160	⊢
	: , :
143	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.zero_is_complex
144	theorem		⊢
	proveit.numbers.addition.elim_zero_any
145	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
146	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
147	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
148	instantiation	161	⊢
	: , : , :
149	instantiation	162, 173	⊢
	:
150	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
151	instantiation	182	⊢
	:
152	theorem		⊢
	proveit.numbers.number_sets.integers.interval_upper_bound
153	instantiation	163, 188, 196	⊢
	: , :
154	assumption		⊢
155	instantiation	163, 164, 165	⊢
	: , :
156	assumption		⊢
157	theorem		⊢
	proveit.numbers.number_sets.integers.nonpos_int_is_int_nonpos
158	instantiation	166, 179, 178, 181, 167, 168*, 169*	⊢
	: , : , :
159	theorem		⊢
	proveit.numbers.negation.double_negation
160	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
161	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
162	theorem		⊢
	proveit.numbers.negation.complex_closure
163	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
164	instantiation	170, 190, 188	⊢
	: , :
165	theorem		⊢
	proveit.numbers.number_sets.integers.zero_is_int
166	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
167	instantiation	171, 199	⊢
	:
168	instantiation	172, 173, 174	⊢
	: , :
169	instantiation	175, 176, 177	⊢
	: , :
170	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
171	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_lower_bound
172	theorem		⊢
	proveit.numbers.addition.commutation
173	instantiation	197, 180, 178	⊢
	: , : , :
174	instantiation	197, 180, 179	⊢
	: , : , :
175	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_basic
176	instantiation	197, 180, 181	⊢
	: , : , :
177	instantiation	182	⊢
	:
178	instantiation	197, 184, 183	⊢
	: , : , :
179	instantiation	197, 184, 185	⊢
	: , : , :
180	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
181	instantiation	186, 187, 199	⊢
	: , : , :
182	axiom		⊢
	proveit.logic.equality.equals_reflexivity
183	instantiation	197, 189, 188	⊢
	: , : , :
184	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
185	instantiation	197, 189, 190	⊢
	: , : , :
186	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
187	instantiation	191, 192	⊢
	: , :
188	instantiation	197, 193, 194	⊢
	: , : , :
189	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
190	instantiation	195, 196	⊢
	:
191	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
192	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
193	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
194	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
195	theorem		⊢
	proveit.numbers.negation.int_closure
196	instantiation	197, 198, 199	⊢
	: , : , :
197	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
198	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
199	assumption		⊢
*equality replacement requirements