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, 230, 139, 141	⊢
	: , : , : , : , :
6	instantiation	12, 230, 139, 141	⊢
	: , : , : , : , :
7	theorem		⊢
	proveit.logic.sets.functions.injections.subset_injection
8	instantiation	13, 133, 41, 228, 14	⊢
	: , : , : , :
9	instantiation	15, 16	⊢
	: , :
10	theorem		⊢
	proveit.physics.quantum.QPE._outcome_prob
11	instantiation	17, 101, 18	,  ⊢
	: , :
12	theorem		⊢
	proveit.core_expr_types.conditionals.true_condition_elimination
13	theorem		⊢
	proveit.numbers.number_sets.integers.interval_subset_eq
14	instantiation	19, 153, 20, 21, 22, 23	⊢
	: , :
15	theorem		⊢
	proveit.logic.booleans.conjunction.left_from_and
16	instantiation	24, 25	⊢
	: , : , :
17	theorem		⊢
	proveit.physics.quantum.QPE._mod_add_closure
18	instantiation	236, 26, 27	,  ⊢
	: , : , :
19	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_all
20	instantiation	164	⊢
	: , : , :
21	instantiation	28, 29, 30	⊢
	: , :
22	instantiation	31, 32, 179, 33, 34, 35*, 36*	⊢
	: , : , :
23	instantiation	37, 38	⊢
	: , :
24	theorem		⊢
	proveit.logic.sets.functions.bijections.membership_unfolding
25	instantiation	39, 40	⊢
	: , : , :
26	instantiation	220, 133, 41	⊢
	: , :
27	assumption		⊢
28	theorem		⊢
	proveit.numbers.ordering.relax_equal_to_less_eq
29	instantiation	236, 223, 42	⊢
	: , : , :
30	instantiation	142	⊢
	:
31	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
32	instantiation	43, 153, 44, 45, 206, 135	⊢
	: , :
33	instantiation	236, 223, 46	⊢
	: , : , :
34	instantiation	47, 222, 221, 212	⊢
	: , : , :
35	instantiation	148, 48, 49, 50	⊢
	: , : , : , :
36	instantiation	160, 51	⊢
	: , :
37	theorem		⊢
	proveit.numbers.ordering.relax_less
38	instantiation	52, 53, 54	⊢
	: , : , :
39	theorem		⊢
	proveit.logic.sets.functions.bijections.elim_domain_condition
40	modus ponens	55, 56	⊢
41	instantiation	234, 57	⊢
	:
42	instantiation	236, 231, 133	⊢
	: , : , :
43	theorem		⊢
	proveit.numbers.addition.add_real_closure
44	instantiation	164	⊢
	: , : , :
45	instantiation	236, 223, 58	⊢
	: , : , :
46	instantiation	236, 231, 221	⊢
	: , : , :
47	theorem		⊢
	proveit.numbers.number_sets.integers.interval_upper_bound
48	instantiation	123, 59, 60	⊢
	: , : , :
49	instantiation	142	⊢
	:
50	instantiation	160, 61	⊢
	: , :
51	instantiation	148, 62, 63, 64	⊢
	: , : , : , :
52	axiom		⊢
	proveit.numbers.ordering.transitivity_less_less
53	instantiation	65, 66	⊢
	:
54	instantiation	67, 233	⊢
	:
55	instantiation	68, 69*	⊢
	: , : , : , : , : , :
56	instantiation	70, 71, 72	⊢
	: , :
57	instantiation	227, 200, 222	⊢
	: , :
58	instantiation	236, 231, 146	⊢
	: , : , :
59	instantiation	162, 107	⊢
	: , : , :
60	instantiation	123, 73, 74	⊢
	: , : , :
61	instantiation	162, 122	⊢
	: , : , :
62	instantiation	75, 167, 195	⊢
	: , :
63	instantiation	160, 76	⊢
	: , :
64	instantiation	160, 77	⊢
	: , :
65	theorem		⊢
	proveit.numbers.number_sets.integers.negative_if_in_neg_int
66	instantiation	78, 79	⊢
	:
67	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
68	theorem		⊢
	proveit.logic.sets.functions.bijections.bijection_transitivity
69	instantiation	162, 80	⊢
	: , : , :
70	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
71	instantiation	81, 82, 133, 228, 101	⊢
	: , : , : , : , :
72	theorem		⊢
	proveit.physics.quantum.QPE._sample_space_bijection
73	instantiation	138, 230, 153, 139, 154, 141, 167, 155, 195, 156	⊢
	: , : , : , : , : , :
74	instantiation	126, 139, 238, 141, 83, 167, 155, 195, 120	⊢
	: , : , : , : , : , : , : , :
75	theorem		⊢
	proveit.numbers.negation.distribute_neg_through_binary_sum
76	instantiation	84, 105, 167, 171, 85	⊢
	: , : , :
77	instantiation	123, 86, 87	⊢
	: , : , :
78	theorem		⊢
	proveit.numbers.negation.int_neg_closure
79	instantiation	88, 89, 175	⊢
	: , :
80	instantiation	160, 90	⊢
	: , :
81	theorem		⊢
	proveit.numbers.modular.interval_left_shift_bijection
82	instantiation	91, 238, 219	⊢
	: , :
83	instantiation	157	⊢
	: , :
84	theorem		⊢
	proveit.numbers.addition.subtraction.subtract_from_add
85	instantiation	129, 92, 93	⊢
	: , : , :
86	instantiation	123, 94, 95	⊢
	: , : , :
87	instantiation	123, 96, 97	⊢
	: , : , :
88	theorem		⊢
	proveit.numbers.addition.add_nat_pos_closure_bin
89	instantiation	98, 200, 99	⊢
	:
90	instantiation	100, 101, 102	⊢
	: , :
91	theorem		⊢
	proveit.numbers.exponentiation.exp_natpos_closure
92	instantiation	123, 103, 104	⊢
	: , : , :
93	instantiation	169, 167, 105	⊢
	: , :
94	instantiation	162, 106	⊢
	: , : , :
95	instantiation	162, 107	⊢
	: , : , :
96	instantiation	123, 108, 109	⊢
	: , : , :
97	instantiation	110, 139, 238, 230, 141, 111, 145, 195, 156, 112*	⊢
	: , : , : , : , : , :
98	theorem		⊢
	proveit.numbers.number_sets.integers.pos_int_is_natural_pos
99	instantiation	113, 114, 115	⊢
	: , : , :
100	axiom		⊢
	proveit.physics.quantum.QPE._mod_add_def
101	theorem		⊢
	proveit.physics.quantum.QPE._best_floor_is_int
102	instantiation	236, 116, 117	⊢
	: , : , :
103	instantiation	138, 230, 238, 139, 118, 141, 167, 156, 171	⊢
	: , : , : , : , : , :
104	instantiation	119, 167, 171, 120	⊢
	: , : , :
105	instantiation	236, 216, 121	⊢
	: , : , :
106	instantiation	162, 136	⊢
	: , : , :
107	instantiation	162, 122	⊢
	: , : , :
108	instantiation	123, 124, 125	⊢
	: , : , :
109	instantiation	126, 139, 230, 238, 141, 127, 165, 145, 195, 156, 128	⊢
	: , : , : , : , : , : , : , :
110	theorem		⊢
	proveit.numbers.addition.association
111	instantiation	157	⊢
	: , :
112	instantiation	129, 130, 131	⊢
	: , : , :
113	theorem		⊢
	proveit.numbers.ordering.transitivity_less_less_eq
114	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
115	instantiation	132, 222, 221, 212	⊢
	: , : , :
116	instantiation	220, 133, 228	⊢
	: , :
117	assumption		⊢
118	instantiation	157	⊢
	: , :
119	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_12
120	instantiation	142	⊢
	:
121	instantiation	134, 135, 181	⊢
	: , :
122	instantiation	162, 136	⊢
	: , : , :
123	axiom		⊢
	proveit.logic.equality.equals_transitivity
124	instantiation	138, 139, 238, 230, 141, 140, 165, 145, 137	⊢
	: , : , : , : , : , :
125	instantiation	138, 238, 153, 139, 140, 154, 141, 165, 145, 155, 195, 156	⊢
	: , : , : , : , : , :
126	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_general
127	instantiation	157	⊢
	: , :
128	instantiation	142	⊢
	:
129	theorem		⊢
	proveit.logic.equality.sub_right_side_into
130	instantiation	143, 195, 209, 144	⊢
	: , : , :
131	instantiation	169, 195, 145	⊢
	: , :
132	theorem		⊢
	proveit.numbers.number_sets.integers.interval_lower_bound
133	instantiation	227, 146, 222	⊢
	: , :
134	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
135	instantiation	147, 179	⊢
	:
136	instantiation	148, 149, 150, 151	⊢
	: , : , : , :
137	instantiation	152, 153, 154, 155, 195, 156	⊢
	: , :
138	theorem		⊢
	proveit.numbers.addition.disassociation
139	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
140	instantiation	157	⊢
	: , :
141	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
142	axiom		⊢
	proveit.logic.equality.equals_reflexivity
143	theorem		⊢
	proveit.numbers.addition.subtraction.subtract_from_add_reversed
144	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
145	instantiation	236, 216, 158	⊢
	: , : , :
146	instantiation	234, 228	⊢
	:
147	theorem		⊢
	proveit.numbers.negation.real_closure
148	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
149	instantiation	162, 159	⊢
	: , : , :
150	instantiation	160, 161	⊢
	: , :
151	instantiation	162, 163	⊢
	: , : , :
152	theorem		⊢
	proveit.numbers.addition.add_complex_closure
153	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
154	instantiation	164	⊢
	: , : , :
155	instantiation	166, 165	⊢
	:
156	instantiation	166, 167	⊢
	:
157	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
158	instantiation	236, 223, 168	⊢
	: , : , :
159	instantiation	169, 170, 171	⊢
	: , :
160	theorem		⊢
	proveit.logic.equality.equals_reversal
161	instantiation	172, 209, 181, 180, 197	⊢
	: , : , :
162	axiom		⊢
	proveit.logic.equality.substitution
163	instantiation	173, 174, 175	⊢
	: , :
164	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
165	instantiation	176, 177, 178	⊢
	: , :
166	theorem		⊢
	proveit.numbers.negation.complex_closure
167	instantiation	236, 216, 179	⊢
	: , : , :
168	instantiation	236, 231, 229	⊢
	: , : , :
169	theorem		⊢
	proveit.numbers.addition.commutation
170	instantiation	236, 216, 180	⊢
	: , : , :
171	instantiation	236, 216, 181	⊢
	: , : , :
172	theorem		⊢
	proveit.numbers.exponentiation.product_of_real_powers
173	theorem		⊢
	proveit.numbers.exponentiation.neg_power_as_div
174	instantiation	236, 182, 183	⊢
	: , : , :
175	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
176	theorem		⊢
	proveit.numbers.multiplication.mult_complex_closure_bin
177	instantiation	184, 195, 185, 186	⊢
	: , :
178	instantiation	236, 216, 187	⊢
	: , : , :
179	instantiation	236, 223, 188	⊢
	: , : , :
180	instantiation	189, 190, 226	⊢
	: , : , :
181	instantiation	236, 223, 191	⊢
	: , : , :
182	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
183	instantiation	236, 192, 193	⊢
	: , : , :
184	theorem		⊢
	proveit.numbers.division.div_complex_closure
185	instantiation	194, 209, 195	⊢
	: , :
186	instantiation	196, 197, 198	⊢
	: , : , :
187	instantiation	236, 223, 199	⊢
	: , : , :
188	instantiation	236, 231, 200	⊢
	: , : , :
189	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
190	instantiation	201, 202	⊢
	: , :
191	instantiation	236, 231, 203	⊢
	: , : , :
192	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
193	instantiation	236, 204, 205	⊢
	: , : , :
194	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
195	instantiation	236, 216, 206	⊢
	: , : , :
196	theorem		⊢
	proveit.logic.equality.sub_left_side_into
197	instantiation	207, 214	⊢
	:
198	instantiation	208, 209	⊢
	:
199	instantiation	236, 231, 210	⊢
	: , : , :
200	instantiation	236, 211, 212	⊢
	: , : , :
201	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
202	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
203	instantiation	234, 222	⊢
	:
204	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
205	instantiation	236, 213, 214	⊢
	: , : , :
206	instantiation	236, 223, 215	⊢
	: , : , :
207	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
208	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
209	instantiation	236, 216, 217	⊢
	: , : , :
210	instantiation	218, 235, 219	⊢
	: , :
211	instantiation	220, 222, 221	⊢
	: , :
212	assumption		⊢
213	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
214	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
215	instantiation	236, 231, 222	⊢
	: , : , :
216	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
217	instantiation	236, 223, 224	⊢
	: , : , :
218	theorem		⊢
	proveit.numbers.exponentiation.exp_int_closure
219	instantiation	236, 225, 226	⊢
	: , : , :
220	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
221	instantiation	227, 228, 229	⊢
	: , :
222	instantiation	236, 237, 230	⊢
	: , : , :
223	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
224	instantiation	236, 231, 235	⊢
	: , : , :
225	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
226	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
227	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
228	instantiation	236, 232, 233	⊢
	: , : , :
229	instantiation	234, 235	⊢
	:
230	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
231	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
232	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
233	theorem		⊢
	proveit.physics.quantum.QPE._two_pow_t_minus_one_is_nat_pos
234	theorem		⊢
	proveit.numbers.negation.int_closure
235	instantiation	236, 237, 238	⊢
	: , : , :
236	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
237	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
238	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
*equality replacement requirements