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	121	⊢
3	instantiation	35, 22, 81	⊢
	: , :
4	instantiation	11, 6, 7	⊢
	: , : , :
5	reference	94	⊢
6	instantiation	8, 172, 18, 9, 10*	⊢
	: , :
7	instantiation	11, 12, 13	⊢
	: , : , :
8	theorem		⊢
	proveit.numbers.rounding.ceil_of_real_above_int
9	instantiation	14, 155, 33, 15, 147, 16*	⊢
	: , : , :
10	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
11	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
12	instantiation	17, 18, 128, 19	⊢
	: , :
13	instantiation	20, 81, 21, 22, 23, 24*	⊢
	: , : , :
14	theorem		⊢
	proveit.numbers.logarithms.log_increasing_less
15	instantiation	25, 121, 26, 27, 28, 29*, 30*	⊢
	: , : , :
16	instantiation	31, 155	⊢
	:
17	theorem		⊢
	proveit.numbers.rounding.ceil_increasing_less_eq
18	instantiation	133, 155, 33, 135	⊢
	: , :
19	instantiation	32, 155, 33, 134, 34, 147	⊢
	: , : , :
20	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
21	instantiation	35, 102, 82	⊢
	: , :
22	instantiation	116, 117, 36	⊢
	: , : , :
23	axiom		⊢
	proveit.physics.quantum.QPE._t_req
24	instantiation	97, 37, 38	⊢
	: , : , :
25	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
26	instantiation	39, 104, 166	⊢
	: , :
27	instantiation	173, 169, 40	⊢
	: , : , :
28	instantiation	41, 104, 166, 167, 42, 43	⊢
	: , : , :
29	instantiation	97, 44, 45	⊢
	: , : , :
30	instantiation	97, 46, 47	⊢
	: , : , :
31	theorem		⊢
	proveit.numbers.logarithms.log_eq_1
32	theorem		⊢
	proveit.numbers.logarithms.log_increasing_less_eq
33	instantiation	142, 48, 155, 72	⊢
	: , :
34	instantiation	49, 121, 50, 51, 52, 53*	⊢
	: , : , :
35	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
36	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
37	instantiation	54, 55, 146, 175, 56, 57, 60, 61, 58	⊢
	: , : , : , : , : , :
38	instantiation	59, 60, 61, 62	⊢
	: , : , :
39	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
40	instantiation	63, 120, 170	⊢
	: , :
41	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_right_factor_bound
42	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
43	instantiation	64, 130	⊢
	:
44	instantiation	67, 65	⊢
	: , : , :
45	instantiation	66, 110	⊢
	:
46	instantiation	67, 68	⊢
	: , : , :
47	instantiation	77, 172, 137, 78*, 69*, 95*	⊢
	: , : , : , :
48	instantiation	173, 157, 70	⊢
	: , : , :
49	theorem		⊢
	proveit.numbers.addition.weak_bound_via_right_term_bound
50	instantiation	71, 167, 121, 72	⊢
	: , :
51	instantiation	173, 73, 139	⊢
	: , : , :
52	instantiation	74, 75, 153, 155, 76	⊢
	: , : , :
53	instantiation	77, 137, 172, 78*, 79*, 80*	⊢
	: , : , : , :
54	theorem		⊢
	proveit.numbers.addition.disassociation
55	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
56	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
57	instantiation	122	⊢
	: , :
58	instantiation	173, 132, 81	⊢
	: , : , :
59	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
60	instantiation	173, 132, 102	⊢
	: , : , :
61	instantiation	173, 132, 82	⊢
	: , : , :
62	instantiation	83	⊢
	:
63	theorem		⊢
	proveit.numbers.multiplication.mult_rational_closure_bin
64	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
65	instantiation	84, 85	⊢
	:
66	theorem		⊢
	proveit.numbers.addition.elim_zero_left
67	axiom		⊢
	proveit.logic.equality.substitution
68	instantiation	123, 85	⊢
	:
69	instantiation	97, 86, 87	⊢
	: , : , :
70	instantiation	173, 162, 88	⊢
	: , : , :
71	theorem		⊢
	proveit.numbers.division.div_real_closure
72	instantiation	89, 163	⊢
	:
73	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
74	theorem		⊢
	proveit.numbers.division.weak_div_from_denom_bound__all_pos
75	instantiation	173, 90, 91	⊢
	: , : , :
76	instantiation	92, 121, 160, 167, 93, 94, 95*	⊢
	: , : , :
77	theorem		⊢
	proveit.numbers.addition.rational_pair_addition
78	instantiation	96, 110	⊢
	:
79	instantiation	97, 98, 99	⊢
	: , : , :
80	instantiation	100, 110	⊢
	:
81	instantiation	101, 102	⊢
	:
82	instantiation	173, 169, 103	⊢
	: , : , :
83	axiom		⊢
	proveit.logic.equality.equals_reflexivity
84	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
85	instantiation	173, 132, 104	⊢
	: , : , :
86	instantiation	111, 146, 105, 106, 115, 114	⊢
	: , : , : , :
87	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_4
88	theorem		⊢
	proveit.numbers.numerals.decimals.posnat5
89	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
90	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
91	instantiation	173, 107, 175	⊢
	: , : , :
92	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
93	instantiation	108, 166, 167, 168	⊢
	: , : , :
94	instantiation	109, 146	⊢
	:
95	instantiation	123, 110	⊢
	:
96	theorem		⊢
	proveit.numbers.division.frac_one_denom
97	axiom		⊢
	proveit.logic.equality.equals_transitivity
98	instantiation	111, 146, 112, 113, 114, 115	⊢
	: , : , : , :
99	theorem		⊢
	proveit.numbers.numerals.decimals.add_4_1
100	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
101	theorem		⊢
	proveit.numbers.negation.real_closure
102	instantiation	116, 117, 118	⊢
	: , : , :
103	instantiation	173, 171, 119	⊢
	: , : , :
104	instantiation	173, 169, 120	⊢
	: , : , :
105	instantiation	122	⊢
	: , :
106	instantiation	122	⊢
	: , :
107	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
108	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_upper_bound
109	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_lower_bound
110	instantiation	173, 132, 121	⊢
	: , : , :
111	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
112	instantiation	122	⊢
	: , :
113	instantiation	122	⊢
	: , :
114	theorem		⊢
	proveit.numbers.numerals.decimals.mult_2_2
115	instantiation	123, 124	⊢
	:
116	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
117	instantiation	125, 126	⊢
	: , :
118	axiom		⊢
	proveit.physics.quantum.QPE._n_in_natural_pos
119	instantiation	127, 128	⊢
	:
120	instantiation	173, 129, 130	⊢
	: , : , :
121	instantiation	173, 169, 131	⊢
	: , : , :
122	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
123	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
124	instantiation	173, 132, 167	⊢
	: , : , :
125	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
126	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
127	axiom		⊢
	proveit.numbers.rounding.ceil_is_an_int
128	instantiation	133, 155, 134, 135	⊢
	: , :
129	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
130	instantiation	136, 148, 158	⊢
	: , :
131	instantiation	173, 171, 137	⊢
	: , : , :
132	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
133	theorem		⊢
	proveit.numbers.logarithms.log_real_pos_real_closure
134	instantiation	138, 155, 139	⊢
	: , :
135	instantiation	140, 141	⊢
	: , :
136	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
137	instantiation	173, 174, 146	⊢
	: , : , :
138	theorem		⊢
	proveit.numbers.addition.add_real_pos_closure_bin
139	instantiation	142, 143, 153, 144	⊢
	: , :
140	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
141	instantiation	145, 175, 146, 147	⊢
	: , :
142	theorem		⊢
	proveit.numbers.division.div_real_pos_closure
143	instantiation	173, 157, 148	⊢
	: , : , :
144	instantiation	149, 150	⊢
	:
145	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq_nat
146	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
147	theorem		⊢
	proveit.numbers.numerals.decimals.less_1_2
148	instantiation	173, 162, 151	⊢
	: , : , :
149	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
150	instantiation	173, 152, 153	⊢
	: , : , :
151	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
152	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
153	instantiation	154, 155, 156	⊢
	: , :
154	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
155	instantiation	173, 157, 158	⊢
	: , : , :
156	instantiation	159, 160, 161	⊢
	:
157	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
158	instantiation	173, 162, 163	⊢
	: , : , :
159	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pos_real_is_real_pos
160	instantiation	164, 166, 167, 168	⊢
	: , : , :
161	instantiation	165, 166, 167, 168	⊢
	: , : , :
162	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
163	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
164	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
165	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
166	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
167	instantiation	173, 169, 170	⊢
	: , : , :
168	axiom		⊢
	proveit.physics.quantum.QPE._eps_in_interval
169	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
170	instantiation	173, 171, 172	⊢
	: , : , :
171	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
172	instantiation	173, 174, 175	⊢
	: , : , :
173	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
174	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
175	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements