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.addition.weak_bound_via_left_term_bound
2	modus ponens	6, 7	⊢
3	modus ponens	8, 9	⊢
4	modus ponens	10, 11	⊢
5	modus ponens	12, 13	⊢
6	instantiation	16	⊢
	: , : , :
7	generalization	14	⊢
8	instantiation	16	⊢
	: , : , :
9	generalization	15	⊢
10	instantiation	16	⊢
	: , : , :
11	generalization	17	⊢
12	instantiation	18	⊢
	: , : , :
13	generalization	19	⊢
14	instantiation	24, 163, 20, 21	,  ⊢
	: , :
15	instantiation	24, 163, 22, 23	,  ⊢
	: , :
16	theorem		⊢
	proveit.numbers.summation.summation_real_closure
17	instantiation	24, 163, 25, 26	,  ⊢
	: , :
18	theorem		⊢
	proveit.numbers.summation.weak_summation_from_summands_bound
19	instantiation	27, 28, 43, 41, 29	,  ⊢
	: , : , :
20	instantiation	32, 68, 186	,  ⊢
	: , :
21	instantiation	33, 30	,  ⊢
	:
22	instantiation	32, 57, 186	,  ⊢
	: , :
23	instantiation	33, 31	,  ⊢
	:
24	theorem		⊢
	proveit.numbers.division.div_real_closure
25	instantiation	32, 78, 186	,  ⊢
	: , :
26	instantiation	33, 34	,  ⊢
	:
27	theorem		⊢
	proveit.numbers.division.weak_div_from_denom_bound__all_pos
28	instantiation	184, 35, 36	⊢
	: , : , :
29	instantiation	37, 134, 57, 78, 38, 39	,  ⊢
	: , : , :
30	instantiation	184, 42, 40	,  ⊢
	: , : , :
31	instantiation	184, 42, 41	,  ⊢
	: , : , :
32	theorem		⊢
	proveit.numbers.exponentiation.exp_real_closure_nat_power
33	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
34	instantiation	184, 42, 43	,  ⊢
	: , : , :
35	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
36	instantiation	184, 44, 179	⊢
	: , : , :
37	theorem		⊢
	proveit.numbers.exponentiation.exp_even_neg_base_lesseq
38	instantiation	45, 46, 84	,  ⊢
	: , :
39	instantiation	47	⊢
	:
40	instantiation	50, 68, 48	,  ⊢
	:
41	instantiation	50, 57, 49	,  ⊢
	:
42	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
43	instantiation	50, 78, 51	,  ⊢
	:
44	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
45	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
46	instantiation	52, 78, 81, 128, 53, 54*	,  ⊢
	: , : , :
47	axiom		⊢
	proveit.logic.equality.equals_reflexivity
48	instantiation	55, 56	,  ⊢
	: , :
49	instantiation	67, 57, 128, 58	,  ⊢
	: , :
50	theorem		⊢
	proveit.numbers.exponentiation.sqrd_pos_closure
51	instantiation	59, 109, 60, 84	,  ⊢
	: , :
52	theorem		⊢
	proveit.numbers.addition.weak_bound_via_right_term_bound
53	instantiation	61, 62, 128, 94, 85, 63, 64*, 65*	⊢
	: , : , :
54	instantiation	153, 66	,  ⊢
	:
55	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
56	instantiation	67, 128, 68, 69	,  ⊢
	: , :
57	instantiation	79, 78, 81	,  ⊢
	: , :
58	instantiation	70, 71	,  ⊢
	: , :
59	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq_int
60	theorem		⊢
	proveit.numbers.number_sets.integers.zero_is_int
61	theorem		⊢
	proveit.numbers.multiplication.reversed_weak_bound_via_right_factor_bound
62	instantiation	93, 163	⊢
	:
63	instantiation	72, 73	⊢
	: , :
64	instantiation	74, 75, 76	⊢
	: , : , :
65	instantiation	77, 88	⊢
	:
66	instantiation	184, 162, 78	,  ⊢
	: , : , :
67	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq
68	instantiation	79, 80, 81	,  ⊢
	: , :
69	instantiation	82, 83	,  ⊢
	: , :
70	theorem		⊢
	proveit.numbers.addition.subtraction.neg_difference
71	instantiation	164, 84, 85	,  ⊢
	: , : , :
72	theorem		⊢
	proveit.numbers.ordering.relax_less
73	instantiation	117, 86	⊢
	:
74	axiom		⊢
	proveit.logic.equality.equals_transitivity
75	instantiation	87, 179, 186, 104, 106, 105, 88, 107, 108	⊢
	: , : , : , : , : , :
76	instantiation	89, 104, 186, 105, 106, 155, 107, 108, 90*	⊢
	: , : , : , : , :
77	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
78	instantiation	184, 168, 91	,  ⊢
	: , : , :
79	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
80	instantiation	184, 168, 92	,  ⊢
	: , : , :
81	instantiation	93, 94	⊢
	:
82	theorem		⊢
	proveit.numbers.addition.subtraction.pos_difference
83	instantiation	95, 96, 97	,  ⊢
	: , : , :
84	instantiation	98, 99, 100	,  ⊢
	: , : , :
85	instantiation	101, 128, 163, 113	⊢
	: , : , :
86	instantiation	132, 146	⊢
	:
87	theorem		⊢
	proveit.numbers.multiplication.disassociation
88	instantiation	102, 155	⊢
	:
89	theorem		⊢
	proveit.numbers.multiplication.mult_neg_any
90	instantiation	103, 104, 186, 105, 106, 107, 108	⊢
	: , : , : , :
91	instantiation	184, 174, 109	,  ⊢
	: , : , :
92	instantiation	184, 174, 110	,  ⊢
	: , : , :
93	theorem		⊢
	proveit.numbers.negation.real_closure
94	instantiation	111, 128, 163, 113	⊢
	: , : , :
95	axiom		⊢
	proveit.numbers.ordering.transitivity_less_less
96	instantiation	112, 128, 163, 113	⊢
	: , : , :
97	instantiation	164, 114, 115	,  ⊢
	: , : , :
98	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less
99	instantiation	116, 138, 139, 125	,  ⊢
	: , : , :
100	instantiation	117, 118	⊢
	:
101	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_co_lower_bound
102	theorem		⊢
	proveit.numbers.negation.complex_closure
103	theorem		⊢
	proveit.numbers.multiplication.elim_one_any
104	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
105	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
106	instantiation	119	⊢
	: , :
107	instantiation	120, 121, 122	⊢
	: , :
108	instantiation	184, 162, 123	⊢
	: , : , :
109	instantiation	184, 124, 125	,  ⊢
	: , : , :
110	instantiation	184, 126, 131	,  ⊢
	: , : , :
111	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_co__is__real
112	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_co_upper_bound
113	theorem		⊢
	proveit.physics.quantum.QPE._scaled_delta_b_floor_in_interval
114	instantiation	127, 163, 128, 129, 158, 130*	⊢
	: , : , :
115	instantiation	170, 152, 177, 131	,  ⊢
	: , : , :
116	theorem		⊢
	proveit.numbers.number_sets.integers.interval_upper_bound
117	theorem		⊢
	proveit.numbers.number_sets.integers.negative_if_in_neg_int
118	instantiation	132, 133	⊢
	:
119	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
120	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
121	instantiation	184, 162, 134	⊢
	: , : , :
122	instantiation	184, 162, 135	⊢
	: , : , :
123	instantiation	136, 137	⊢
	:
124	instantiation	172, 138, 139	⊢
	: , :
125	assumption		⊢
126	instantiation	172, 152, 177	⊢
	: , :
127	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
128	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
129	instantiation	184, 168, 140	⊢
	: , : , :
130	instantiation	141, 142, 143	⊢
	: , : , :
131	assumption		⊢
132	theorem		⊢
	proveit.numbers.negation.int_neg_closure
133	instantiation	144, 145, 146	⊢
	: , :
134	instantiation	184, 168, 147	⊢
	: , : , :
135	instantiation	148, 149, 150	⊢
	: , : , :
136	theorem		⊢
	proveit.physics.quantum.QPE._delta_b_is_real
137	theorem		⊢
	proveit.physics.quantum.QPE._best_floor_is_int
138	instantiation	176, 151, 175	⊢
	: , :
139	instantiation	182, 152	⊢
	:
140	instantiation	184, 174, 161	⊢
	: , : , :
141	theorem		⊢
	proveit.logic.equality.sub_right_side_into
142	instantiation	153, 155	⊢
	:
143	instantiation	154, 155, 156	⊢
	: , :
144	theorem		⊢
	proveit.numbers.addition.add_nat_pos_closure_bin
145	instantiation	157, 161, 158	⊢
	:
146	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
147	instantiation	184, 174, 183	⊢
	: , : , :
148	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
149	instantiation	159, 160	⊢
	: , :
150	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
151	instantiation	182, 177	⊢
	:
152	instantiation	176, 161, 175	⊢
	: , :
153	theorem		⊢
	proveit.numbers.addition.elim_zero_right
154	theorem		⊢
	proveit.numbers.addition.commutation
155	instantiation	184, 162, 163	⊢
	: , : , :
156	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.zero_is_complex
157	theorem		⊢
	proveit.numbers.number_sets.integers.pos_int_is_natural_pos
158	instantiation	164, 165, 166	⊢
	: , : , :
159	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
160	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
161	instantiation	184, 167, 171	⊢
	: , : , :
162	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
163	instantiation	184, 168, 169	⊢
	: , : , :
164	theorem		⊢
	proveit.numbers.ordering.transitivity_less_less_eq
165	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
166	instantiation	170, 175, 173, 171	⊢
	: , : , :
167	instantiation	172, 175, 173	⊢
	: , :
168	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
169	instantiation	184, 174, 175	⊢
	: , : , :
170	theorem		⊢
	proveit.numbers.number_sets.integers.interval_lower_bound
171	assumption		⊢
172	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
173	instantiation	176, 177, 178	⊢
	: , :
174	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
175	instantiation	184, 185, 179	⊢
	: , : , :
176	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
177	instantiation	184, 180, 181	⊢
	: , : , :
178	instantiation	182, 183	⊢
	:
179	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
180	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
181	theorem		⊢
	proveit.physics.quantum.QPE._two_pow_t_minus_one_is_nat_pos
182	theorem		⊢
	proveit.numbers.negation.int_closure
183	instantiation	184, 185, 186	⊢
	: , : , :
184	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
185	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
186	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
*equality replacement requirements