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	⊢
	: , :
1	theorem		⊢
	proveit.numbers.ordering.relax_less
2	instantiation	106, 3, 4	⊢
	: , : , :
3	instantiation	5, 6, 7	⊢
	: , : , :
4	modus ponens	8, 9	⊢
5	theorem		⊢
	proveit.logic.equality.sub_right_side_into
6	instantiation	10, 11, 12	⊢
	: , : , :
7	instantiation	13, 113	⊢
	:
8	instantiation	14, 15, 16	⊢
	: , : , : , : , : , :
9	theorem		⊢
	proveit.physics.quantum.QPE._precision_guarantee_lemma_02
10	theorem		⊢
	proveit.logic.equality.sub_left_side_into
11	instantiation	106, 17, 18	⊢
	: , : , :
12	instantiation	19, 113	⊢
	:
13	axiom		⊢
	proveit.physics.quantum.QPE._success_def
14	theorem		⊢
	proveit.statistics.constrained_event_prob_bound
15	theorem		⊢
	proveit.physics.quantum.QPE._Omega_is_sample_space
16	instantiation	40, 20	⊢
	: , :
17	theorem		⊢
	proveit.physics.quantum.QPE._precision_guarantee_lemma_01
18	instantiation	21, 58, 22, 23, 24, 25*	⊢
	: , : , :
19	theorem		⊢
	proveit.physics.quantum.QPE._success_complements_failure
20	instantiation	26, 27	⊢
	: , : , :
21	theorem		⊢
	proveit.numbers.addition.weak_bound_via_right_term_bound
22	instantiation	43, 48, 49	⊢
	: , :
23	instantiation	53, 29	⊢
	:
24	instantiation	28, 29, 30, 31, 32*	⊢
	: , :
25	instantiation	33, 114, 125, 34, 35, 36, 37, 38, 39	⊢
	: , : , : , : , : , :
26	theorem		⊢
	proveit.logic.sets.functions.injections.membership_unfolding
27	instantiation	40, 41	⊢
	: , :
28	theorem		⊢
	proveit.numbers.negation.negated_weak_bound
29	instantiation	42, 113	⊢
	:
30	instantiation	43, 52, 54	⊢
	: , :
31	instantiation	44, 113	⊢
	:
32	instantiation	45, 46, 47	⊢
	: , :
33	theorem		⊢
	proveit.numbers.addition.disassociation
34	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
35	instantiation	75	⊢
	: , :
36	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
37	instantiation	132, 86, 58	⊢
	: , : , :
38	instantiation	132, 86, 48	⊢
	: , : , :
39	instantiation	132, 86, 49	⊢
	: , : , :
40	theorem		⊢
	proveit.logic.booleans.conjunction.left_from_and
41	instantiation	50, 51	⊢
	: , : , :
42	theorem		⊢
	proveit.physics.quantum.QPE._pfail_in_real
43	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
44	theorem		⊢
	proveit.physics.quantum.QPE._failure_upper_bound
45	theorem		⊢
	proveit.numbers.negation.distribute_neg_through_binary_sum
46	instantiation	132, 86, 52	⊢
	: , : , :
47	instantiation	132, 86, 54	⊢
	: , : , :
48	instantiation	53, 52	⊢
	:
49	instantiation	53, 54	⊢
	:
50	theorem		⊢
	proveit.logic.sets.functions.bijections.membership_unfolding
51	theorem		⊢
	proveit.physics.quantum.QPE._sample_space_bijection
52	instantiation	57, 58, 55, 56	⊢
	: , :
53	theorem		⊢
	proveit.numbers.negation.real_closure
54	instantiation	57, 58, 59, 60	⊢
	: , :
55	instantiation	64, 87, 74	⊢
	: , :
56	instantiation	67, 125, 61, 62, 78	⊢
	: , :
57	theorem		⊢
	proveit.numbers.division.div_real_closure
58	instantiation	132, 93, 63	⊢
	: , : , :
59	instantiation	64, 65, 66	⊢
	: , :
60	instantiation	67, 125, 68, 69, 70	⊢
	: , :
61	instantiation	75	⊢
	: , :
62	instantiation	132, 84, 71	⊢
	: , : , :
63	instantiation	132, 99, 111	⊢
	: , : , :
64	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
65	instantiation	132, 93, 72	⊢
	: , : , :
66	instantiation	73, 74, 125	⊢
	: , :
67	theorem		⊢
	proveit.numbers.multiplication.mult_not_eq_zero
68	instantiation	75	⊢
	: , :
69	instantiation	132, 84, 76	⊢
	: , : , :
70	instantiation	77, 78, 79	⊢
	: , :
71	instantiation	132, 91, 80	⊢
	: , : , :
72	instantiation	132, 99, 81	⊢
	: , : , :
73	theorem		⊢
	proveit.numbers.exponentiation.exp_real_closure_nat_power
74	instantiation	132, 93, 82	⊢
	: , : , :
75	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
76	instantiation	132, 91, 83	⊢
	: , : , :
77	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_nonzero_closure
78	instantiation	132, 84, 85	⊢
	: , : , :
79	instantiation	132, 86, 87	⊢
	: , : , :
80	instantiation	132, 97, 88	⊢
	: , : , :
81	instantiation	132, 124, 89	⊢
	: , : , :
82	instantiation	132, 99, 103	⊢
	: , : , :
83	instantiation	132, 97, 90	⊢
	: , : , :
84	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
85	instantiation	132, 91, 92	⊢
	: , : , :
86	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
87	instantiation	132, 93, 94	⊢
	: , : , :
88	instantiation	132, 100, 95	⊢
	: , : , :
89	theorem		⊢
	proveit.numbers.numerals.decimals.nat4
90	instantiation	132, 100, 96	⊢
	: , : , :
91	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
92	instantiation	132, 97, 98	⊢
	: , : , :
93	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
94	instantiation	132, 99, 120	⊢
	: , : , :
95	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
96	theorem		⊢
	proveit.numbers.numerals.decimals.posnat4
97	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
98	instantiation	132, 100, 101	⊢
	: , : , :
99	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
100	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
101	instantiation	102, 103, 104	⊢
	:
102	theorem		⊢
	proveit.numbers.number_sets.integers.pos_int_is_natural_pos
103	instantiation	132, 105, 113	⊢
	: , : , :
104	instantiation	106, 107, 108	⊢
	: , : , :
105	instantiation	109, 111, 112	⊢
	: , :
106	theorem		⊢
	proveit.numbers.ordering.transitivity_less_less_eq
107	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
108	instantiation	110, 111, 112, 113	⊢
	: , : , :
109	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
110	theorem		⊢
	proveit.numbers.number_sets.integers.interval_lower_bound
111	instantiation	132, 124, 114	⊢
	: , : , :
112	instantiation	126, 115, 116	⊢
	: , :
113	theorem		⊢
	proveit.physics.quantum.QPE._e_value_in_e_domain
114	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
115	instantiation	117, 120, 118	⊢
	: , :
116	instantiation	119, 120	⊢
	:
117	theorem		⊢
	proveit.numbers.exponentiation.exp_int_closure
118	instantiation	121, 122, 123	⊢
	:
119	theorem		⊢
	proveit.numbers.negation.int_closure
120	instantiation	132, 124, 125	⊢
	: , : , :
121	theorem		⊢
	proveit.numbers.number_sets.integers.nonneg_int_is_natural
122	instantiation	126, 127, 128	⊢
	: , :
123	instantiation	129, 130	⊢
	: , :
124	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
125	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
126	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
127	instantiation	132, 131, 136	⊢
	: , : , :
128	instantiation	132, 133, 134	⊢
	: , : , :
129	theorem		⊢
	proveit.numbers.addition.subtraction.nonneg_difference
130	instantiation	135, 136	⊢
	:
131	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
132	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
133	theorem		⊢
	proveit.numbers.number_sets.integers.neg_int_within_int
134	instantiation	137, 138	⊢
	:
135	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_lower_bound
136	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
137	theorem		⊢
	proveit.numbers.negation.int_neg_closure
138	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
*equality replacement requirements