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	⊢
	: , : , :
1	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right
2	reference	116	⊢
3	instantiation	29, 21, 69	⊢
	: , :
4	instantiation	11, 6, 7	⊢
	: , : , :
5	reference	86	⊢
6	instantiation	8, 168, 17, 9, 10*	⊢
	: , :
7	instantiation	11, 12, 13	⊢
	: , : , :
8	theorem		⊢
	proveit.numbers.rounding.ceil_of_real_above_int
9	instantiation	14, 151, 27, 15	⊢
	: , :
10	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
11	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
12	instantiation	16, 17, 120, 18	⊢
	: , :
13	instantiation	19, 69, 20, 21, 22, 23*	⊢
	: , : , :
14	theorem		⊢
	proveit.numbers.logarithms.log_base_large_a_greater_one
15	instantiation	24, 143, 25	⊢
	: , :
16	theorem		⊢
	proveit.numbers.rounding.ceil_increasing_less_eq
17	instantiation	126, 151, 27, 128	⊢
	: , :
18	instantiation	26, 151, 27, 127, 28, 143	⊢
	: , : , :
19	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
20	instantiation	29, 92, 70	⊢
	: , :
21	instantiation	106, 107, 30	⊢
	: , : , :
22	axiom		⊢
	proveit.physics.quantum.QPE._t_req
23	instantiation	97, 31, 32	⊢
	: , : , :
24	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
25	instantiation	33, 116, 34, 35, 36, 37*, 38*	⊢
	: , : , :
26	theorem		⊢
	proveit.numbers.logarithms.log_increasing_less_eq
27	instantiation	169, 153, 39	⊢
	: , : , :
28	instantiation	40, 116, 110, 41, 42, 43*	⊢
	: , : , :
29	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
30	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
31	instantiation	44, 45, 142, 171, 46, 47, 50, 51, 48	⊢
	: , : , : , : , : , :
32	instantiation	49, 50, 51, 52	⊢
	: , : , :
33	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
34	instantiation	53, 110, 162	⊢
	: , :
35	instantiation	169, 165, 54	⊢
	: , : , :
36	instantiation	55, 110, 162, 163, 56, 57	⊢
	: , : , :
37	instantiation	97, 58, 59	⊢
	: , : , :
38	instantiation	97, 60, 61	⊢
	: , : , :
39	instantiation	137, 62, 154	⊢
	: , :
40	theorem		⊢
	proveit.numbers.addition.weak_bound_via_right_term_bound
41	instantiation	169, 63, 134	⊢
	: , : , :
42	instantiation	64, 65, 149, 151, 66	⊢
	: , : , :
43	instantiation	78, 132, 168, 79*, 67*, 68*	⊢
	: , : , : , :
44	theorem		⊢
	proveit.numbers.addition.disassociation
45	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
46	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
47	instantiation	122	⊢
	: , :
48	instantiation	169, 131, 69	⊢
	: , : , :
49	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
50	instantiation	169, 131, 92	⊢
	: , : , :
51	instantiation	169, 131, 70	⊢
	: , : , :
52	instantiation	71	⊢
	:
53	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
54	instantiation	72, 121, 166	⊢
	: , :
55	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_right_factor_bound
56	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
57	instantiation	73, 130	⊢
	:
58	instantiation	76, 74	⊢
	: , : , :
59	instantiation	75, 103	⊢
	:
60	instantiation	76, 77	⊢
	: , : , :
61	instantiation	78, 168, 132, 79*, 80*, 87*	⊢
	: , : , : , :
62	instantiation	169, 158, 81	⊢
	: , : , :
63	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
64	theorem		⊢
	proveit.numbers.division.weak_div_from_denom_bound__all_pos
65	instantiation	169, 82, 83	⊢
	: , : , :
66	instantiation	84, 116, 156, 163, 85, 86, 87*	⊢
	: , : , :
67	instantiation	97, 88, 89	⊢
	: , : , :
68	instantiation	90, 103	⊢
	:
69	instantiation	91, 92	⊢
	:
70	instantiation	169, 165, 93	⊢
	: , : , :
71	axiom		⊢
	proveit.logic.equality.equals_reflexivity
72	theorem		⊢
	proveit.numbers.multiplication.mult_rational_closure_bin
73	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
74	instantiation	94, 95	⊢
	:
75	theorem		⊢
	proveit.numbers.addition.elim_zero_left
76	axiom		⊢
	proveit.logic.equality.substitution
77	instantiation	123, 95	⊢
	:
78	theorem		⊢
	proveit.numbers.addition.rational_pair_addition
79	instantiation	96, 103	⊢
	:
80	instantiation	97, 98, 99	⊢
	: , : , :
81	theorem		⊢
	proveit.numbers.numerals.decimals.posnat5
82	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
83	instantiation	169, 100, 171	⊢
	: , : , :
84	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
85	instantiation	101, 162, 163, 164	⊢
	: , : , :
86	instantiation	102, 142	⊢
	:
87	instantiation	123, 103	⊢
	:
88	instantiation	111, 142, 104, 105, 115, 114	⊢
	: , : , : , :
89	theorem		⊢
	proveit.numbers.numerals.decimals.add_4_1
90	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
91	theorem		⊢
	proveit.numbers.negation.real_closure
92	instantiation	106, 107, 108	⊢
	: , : , :
93	instantiation	169, 167, 109	⊢
	: , : , :
94	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
95	instantiation	169, 131, 110	⊢
	: , : , :
96	theorem		⊢
	proveit.numbers.division.frac_one_denom
97	axiom		⊢
	proveit.logic.equality.equals_transitivity
98	instantiation	111, 142, 112, 113, 114, 115	⊢
	: , : , : , :
99	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_4
100	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
101	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_upper_bound
102	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_lower_bound
103	instantiation	169, 131, 116	⊢
	: , : , :
104	instantiation	122	⊢
	: , :
105	instantiation	122	⊢
	: , :
106	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
107	instantiation	117, 118	⊢
	: , :
108	axiom		⊢
	proveit.physics.quantum.QPE._n_in_natural_pos
109	instantiation	119, 120	⊢
	:
110	instantiation	169, 165, 121	⊢
	: , : , :
111	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
112	instantiation	122	⊢
	: , :
113	instantiation	122	⊢
	: , :
114	instantiation	123, 124	⊢
	:
115	theorem		⊢
	proveit.numbers.numerals.decimals.mult_2_2
116	instantiation	169, 165, 125	⊢
	: , : , :
117	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
118	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
119	axiom		⊢
	proveit.numbers.rounding.ceil_is_an_int
120	instantiation	126, 151, 127, 128	⊢
	: , :
121	instantiation	169, 129, 130	⊢
	: , : , :
122	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
123	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
124	instantiation	169, 131, 163	⊢
	: , : , :
125	instantiation	169, 167, 132	⊢
	: , : , :
126	theorem		⊢
	proveit.numbers.logarithms.log_real_pos_real_closure
127	instantiation	133, 151, 134	⊢
	: , :
128	instantiation	135, 136	⊢
	: , :
129	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
130	instantiation	137, 144, 154	⊢
	: , :
131	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
132	instantiation	169, 170, 142	⊢
	: , : , :
133	theorem		⊢
	proveit.numbers.addition.add_real_pos_closure_bin
134	instantiation	138, 139, 149, 140	⊢
	: , :
135	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
136	instantiation	141, 171, 142, 143	⊢
	: , :
137	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
138	theorem		⊢
	proveit.numbers.division.div_real_pos_closure
139	instantiation	169, 153, 144	⊢
	: , : , :
140	instantiation	145, 146	⊢
	:
141	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq_nat
142	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
143	theorem		⊢
	proveit.numbers.numerals.decimals.less_1_2
144	instantiation	169, 158, 147	⊢
	: , : , :
145	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
146	instantiation	169, 148, 149	⊢
	: , : , :
147	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
148	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
149	instantiation	150, 151, 152	⊢
	: , :
150	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
151	instantiation	169, 153, 154	⊢
	: , : , :
152	instantiation	155, 156, 157	⊢
	:
153	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
154	instantiation	169, 158, 159	⊢
	: , : , :
155	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pos_real_is_real_pos
156	instantiation	160, 162, 163, 164	⊢
	: , : , :
157	instantiation	161, 162, 163, 164	⊢
	: , : , :
158	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
159	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
160	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
161	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
162	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
163	instantiation	169, 165, 166	⊢
	: , : , :
164	axiom		⊢
	proveit.physics.quantum.QPE._eps_in_interval
165	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
166	instantiation	169, 167, 168	⊢
	: , : , :
167	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
168	instantiation	169, 170, 171	⊢
	: , : , :
169	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
170	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
171	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements