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.statistics.prob_of_all_as_sum
2	theorem		⊢
	proveit.physics.quantum.QPE._Omega_is_sample_space
3	instantiation	7, 8, 9	⊢
	: , : , : , :
4	instantiation	10, 11	,  ⊢
	:
5	instantiation	12, 206, 119, 121	⊢
	: , : , : , : , :
6	instantiation	12, 206, 119, 121	⊢
	: , : , : , : , :
7	theorem		⊢
	proveit.logic.sets.functions.injections.subset_injection
8	instantiation	13, 138, 40, 167, 14	⊢
	: , : , : , :
9	instantiation	15, 16	⊢
	: , :
10	theorem		⊢
	proveit.physics.quantum.QPE._outcome_prob
11	instantiation	17, 110, 18	,  ⊢
	: , :
12	theorem		⊢
	proveit.core_expr_types.conditionals.true_condition_elimination
13	theorem		⊢
	proveit.numbers.number_sets.integers.interval_subset_eq
14	instantiation	19, 20, 21, 22, 23, 24	⊢
	: , :
15	theorem		⊢
	proveit.logic.booleans.conjunction.left_from_and
16	instantiation	25, 26	⊢
	: , : , :
17	theorem		⊢
	proveit.physics.quantum.QPE._mod_add_closure
18	instantiation	207, 27, 28	,  ⊢
	: , : , :
19	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_all
20	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
21	instantiation	29	⊢
	: , : , :
22	instantiation	30, 31, 32	⊢
	: , : , :
23	instantiation	41, 172, 50, 33, 34, 35*	⊢
	: , : , :
24	instantiation	36, 42, 37	⊢
	: , :
25	theorem		⊢
	proveit.logic.sets.functions.bijections.membership_unfolding
26	instantiation	38, 39	⊢
	: , : , :
27	instantiation	137, 40, 167	⊢
	: , :
28	assumption		⊢
29	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
30	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
31	instantiation	41, 75, 172, 42, 43, 44*, 45*	⊢
	: , : , :
32	instantiation	46, 47	⊢
	: , :
33	instantiation	48, 51, 172	⊢
	: , :
34	instantiation	49, 50, 51, 172, 52, 53	⊢
	: , : , :
35	instantiation	126, 54, 79, 55	⊢
	: , : , : , :
36	theorem		⊢
	proveit.numbers.ordering.relax_equal_to_less_eq
37	instantiation	108	⊢
	:
38	theorem		⊢
	proveit.logic.sets.functions.bijections.elim_domain_condition
39	modus ponens	56, 57	⊢
40	instantiation	151, 91, 201	⊢
	: , :
41	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
42	instantiation	180, 181, 179	⊢
	: , : , :
43	instantiation	58, 179	⊢
	:
44	instantiation	153, 161, 59	⊢
	: , :
45	instantiation	105, 60, 61	⊢
	: , : , :
46	theorem		⊢
	proveit.numbers.ordering.relax_less
47	instantiation	62, 63	⊢
	:
48	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
49	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right
50	instantiation	207, 196, 64	⊢
	: , : , :
51	instantiation	207, 196, 65	⊢
	: , : , :
52	instantiation	66, 201, 113, 114	⊢
	: , : , :
53	instantiation	67, 206	⊢
	:
54	instantiation	105, 68, 69	⊢
	: , : , :
55	instantiation	140, 115	⊢
	: , :
56	instantiation	70, 71*	⊢
	: , : , : , : , : , :
57	instantiation	72, 73, 74	⊢
	: , :
58	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_lower_bound
59	instantiation	207, 186, 75	⊢
	: , : , :
60	instantiation	105, 76, 77	⊢
	: , : , :
61	instantiation	78, 122, 79	⊢
	: , :
62	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
63	instantiation	80, 81, 159	⊢
	: , :
64	instantiation	207, 204, 91	⊢
	: , : , :
65	instantiation	207, 204, 113	⊢
	: , : , :
66	theorem		⊢
	proveit.numbers.number_sets.integers.interval_upper_bound
67	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_lower_bound
68	instantiation	142, 82	⊢
	: , : , :
69	instantiation	105, 83, 84	⊢
	: , : , :
70	theorem		⊢
	proveit.logic.sets.functions.bijections.bijection_transitivity
71	instantiation	142, 85	⊢
	: , : , :
72	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
73	instantiation	86, 87, 138, 167, 110	⊢
	: , : , : , : , :
74	theorem		⊢
	proveit.physics.quantum.QPE._sample_space_bijection
75	instantiation	207, 196, 88	⊢
	: , : , :
76	instantiation	142, 115	⊢
	: , : , :
77	instantiation	142, 89	⊢
	: , : , :
78	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_basic
79	instantiation	108	⊢
	:
80	theorem		⊢
	proveit.numbers.addition.add_nat_pos_closure_bin
81	instantiation	90, 91, 92	⊢
	:
82	instantiation	105, 93, 94	⊢
	: , : , :
83	instantiation	116, 119, 209, 206, 121, 95, 122, 155, 161	⊢
	: , : , : , : , : , :
84	instantiation	96, 161, 122, 97	⊢
	: , : , :
85	instantiation	140, 98	⊢
	: , :
86	theorem		⊢
	proveit.numbers.modular.interval_left_shift_bijection
87	instantiation	99, 209, 189	⊢
	: , :
88	instantiation	207, 204, 152	⊢
	: , : , :
89	instantiation	142, 115	⊢
	: , : , :
90	theorem		⊢
	proveit.numbers.number_sets.integers.pos_int_is_natural_pos
91	instantiation	207, 100, 114	⊢
	: , : , :
92	instantiation	101, 102, 103	⊢
	: , : , :
93	instantiation	142, 104	⊢
	: , : , :
94	instantiation	105, 106, 107	⊢
	: , : , :
95	instantiation	130	⊢
	: , :
96	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
97	instantiation	108	⊢
	:
98	instantiation	109, 110, 111	⊢
	: , :
99	theorem		⊢
	proveit.numbers.exponentiation.exp_natpos_closure
100	instantiation	137, 201, 113	⊢
	: , :
101	theorem		⊢
	proveit.numbers.ordering.transitivity_less_less_eq
102	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
103	instantiation	112, 201, 113, 114	⊢
	: , : , :
104	instantiation	142, 115	⊢
	: , : , :
105	axiom		⊢
	proveit.logic.equality.equals_transitivity
106	instantiation	116, 119, 209, 206, 121, 117, 122, 150, 161	⊢
	: , : , : , : , : , :
107	instantiation	118, 206, 209, 119, 120, 121, 122, 150, 161, 123*	⊢
	: , : , : , : , : , :
108	axiom		⊢
	proveit.logic.equality.equals_reflexivity
109	axiom		⊢
	proveit.physics.quantum.QPE._mod_add_def
110	theorem		⊢
	proveit.physics.quantum.QPE._best_floor_is_int
111	instantiation	207, 124, 125	⊢
	: , : , :
112	theorem		⊢
	proveit.numbers.number_sets.integers.interval_lower_bound
113	instantiation	151, 167, 190	⊢
	: , :
114	assumption		⊢
115	instantiation	126, 127, 128, 129	⊢
	: , : , : , :
116	theorem		⊢
	proveit.numbers.addition.disassociation
117	instantiation	130	⊢
	: , :
118	theorem		⊢
	proveit.numbers.addition.association
119	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
120	instantiation	130	⊢
	: , :
121	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
122	instantiation	131, 132, 133	⊢
	: , :
123	instantiation	134, 135, 136	⊢
	: , : , :
124	instantiation	137, 138, 167	⊢
	: , :
125	assumption		⊢
126	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
127	instantiation	142, 139	⊢
	: , : , :
128	instantiation	140, 141	⊢
	: , :
129	instantiation	142, 143	⊢
	: , : , :
130	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
131	theorem		⊢
	proveit.numbers.multiplication.mult_complex_closure_bin
132	instantiation	144, 161, 145, 146	⊢
	: , :
133	instantiation	207, 186, 147	⊢
	: , : , :
134	theorem		⊢
	proveit.logic.equality.sub_right_side_into
135	instantiation	148, 161, 175, 149	⊢
	: , : , :
136	instantiation	153, 161, 150	⊢
	: , :
137	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
138	instantiation	151, 152, 201	⊢
	: , :
139	instantiation	153, 154, 155	⊢
	: , :
140	theorem		⊢
	proveit.logic.equality.equals_reversal
141	instantiation	156, 175, 169, 168, 163	⊢
	: , : , :
142	axiom		⊢
	proveit.logic.equality.substitution
143	instantiation	157, 158, 159	⊢
	: , :
144	theorem		⊢
	proveit.numbers.division.div_complex_closure
145	instantiation	160, 175, 161	⊢
	: , :
146	instantiation	162, 163, 164	⊢
	: , : , :
147	instantiation	207, 196, 165	⊢
	: , : , :
148	theorem		⊢
	proveit.numbers.addition.subtraction.subtract_from_add_reversed
149	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
150	instantiation	207, 186, 166	⊢
	: , : , :
151	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
152	instantiation	200, 167	⊢
	:
153	theorem		⊢
	proveit.numbers.addition.commutation
154	instantiation	207, 186, 168	⊢
	: , : , :
155	instantiation	207, 186, 169	⊢
	: , : , :
156	theorem		⊢
	proveit.numbers.exponentiation.product_of_real_powers
157	theorem		⊢
	proveit.numbers.exponentiation.neg_power_as_div
158	instantiation	207, 170, 171	⊢
	: , : , :
159	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
160	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
161	instantiation	207, 186, 172	⊢
	: , : , :
162	theorem		⊢
	proveit.logic.equality.sub_left_side_into
163	instantiation	173, 203	⊢
	:
164	instantiation	174, 175	⊢
	:
165	instantiation	207, 204, 176	⊢
	: , : , :
166	instantiation	207, 196, 177	⊢
	: , : , :
167	instantiation	207, 178, 179	⊢
	: , : , :
168	instantiation	180, 181, 199	⊢
	: , : , :
169	instantiation	207, 196, 182	⊢
	: , : , :
170	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
171	instantiation	207, 183, 184	⊢
	: , : , :
172	instantiation	207, 196, 185	⊢
	: , : , :
173	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
174	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
175	instantiation	207, 186, 187	⊢
	: , : , :
176	instantiation	188, 205, 189	⊢
	: , :
177	instantiation	207, 204, 190	⊢
	: , : , :
178	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
179	theorem		⊢
	proveit.physics.quantum.QPE._two_pow_t_minus_one_is_nat_pos
180	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
181	instantiation	191, 192	⊢
	: , :
182	instantiation	207, 204, 193	⊢
	: , : , :
183	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
184	instantiation	207, 194, 195	⊢
	: , : , :
185	instantiation	207, 204, 201	⊢
	: , : , :
186	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
187	instantiation	207, 196, 197	⊢
	: , : , :
188	theorem		⊢
	proveit.numbers.exponentiation.exp_int_closure
189	instantiation	207, 198, 199	⊢
	: , : , :
190	instantiation	200, 205	⊢
	:
191	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
192	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
193	instantiation	200, 201	⊢
	:
194	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
195	instantiation	207, 202, 203	⊢
	: , : , :
196	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
197	instantiation	207, 204, 205	⊢
	: , : , :
198	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
199	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
200	theorem		⊢
	proveit.numbers.negation.int_closure
201	instantiation	207, 208, 206	⊢
	: , : , :
202	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
203	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
204	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
205	instantiation	207, 208, 209	⊢
	: , : , :
206	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
207	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
208	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
209	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
*equality replacement requirements