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	⊢
	: , : , :
1	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
2	instantiation	6, 7, 4, 8, 5, 11	⊢
	: , : , :
3	instantiation	6, 7, 8, 9, 10, 11	⊢
	: , : , :
4	instantiation	134, 13, 17	⊢
	: , :
5	instantiation	12, 17, 13, 16, 14	⊢
	: , : , :
6	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
7	instantiation	246, 234, 15	⊢
	: , : , :
8	instantiation	134, 16, 17	⊢
	: , :
9	instantiation	134, 16, 18	⊢
	: , :
10	instantiation	90, 16, 17, 18, 19	⊢
	: , : , :
11	instantiation	139, 20	⊢
	: , :
12	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
13	modus ponens	21, 22	⊢
14	modus ponens	23, 24	⊢
15	instantiation	246, 25, 35	⊢
	: , : , :
16	modus ponens	26, 27	⊢
17	modus ponens	28, 29	⊢
18	modus ponens	30, 31	⊢
19	modus ponens	32, 33	⊢
20	instantiation	34, 35	⊢
	:
21	instantiation	40	⊢
	: , : , :
22	generalization	36	⊢
23	instantiation	42	⊢
	: , : , :
24	generalization	37	⊢
25	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
26	instantiation	40	⊢
	: , : , :
27	generalization	38	⊢
28	instantiation	40	⊢
	: , : , :
29	generalization	39	⊢
30	instantiation	40	⊢
	: , : , :
31	generalization	41	⊢
32	instantiation	42	⊢
	: , : , :
33	generalization	43	⊢
34	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
35	instantiation	44, 82, 45	⊢
	: , :
36	instantiation	55, 227, 46, 47	,  ⊢
	: , :
37	instantiation	48, 49, 79, 75, 50	,  ⊢
	: , : , :
38	instantiation	55, 227, 51, 52	,  ⊢
	: , :
39	instantiation	55, 227, 53, 54	,  ⊢
	: , :
40	theorem		⊢
	proveit.numbers.summation.summation_real_closure
41	instantiation	55, 227, 56, 57	,  ⊢
	: , :
42	theorem		⊢
	proveit.numbers.summation.weak_summation_from_summands_bound
43	instantiation	139, 58	,  ⊢
	: , :
44	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
45	instantiation	246, 96, 59	⊢
	: , : , :
46	instantiation	69, 102, 248	,  ⊢
	: , :
47	instantiation	67, 60	,  ⊢
	:
48	theorem		⊢
	proveit.numbers.division.weak_div_from_denom_bound__all_pos
49	instantiation	246, 61, 62	⊢
	: , : , :
50	instantiation	63, 207, 102, 119, 64, 65	,  ⊢
	: , : , :
51	instantiation	69, 119, 248	,  ⊢
	: , :
52	instantiation	67, 66	,  ⊢
	:
53	instantiation	69, 121, 248	,  ⊢
	: , :
54	instantiation	67, 68	,  ⊢
	:
55	theorem		⊢
	proveit.numbers.division.div_real_closure
56	instantiation	69, 86, 248	,  ⊢
	: , :
57	instantiation	70, 97	,  ⊢
	:
58	instantiation	71, 72, 73, 81, 74	,  ⊢
	: , : , :
59	theorem		⊢
	proveit.numbers.numerals.decimals.posnat4
60	instantiation	246, 80, 75	,  ⊢
	: , : , :
61	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
62	instantiation	246, 76, 245	⊢
	: , : , :
63	theorem		⊢
	proveit.numbers.exponentiation.exp_even_neg_base_lesseq
64	instantiation	98, 77, 131	,  ⊢
	: , :
65	instantiation	78	⊢
	:
66	instantiation	246, 80, 79	,  ⊢
	: , : , :
67	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
68	instantiation	246, 80, 81	,  ⊢
	: , : , :
69	theorem		⊢
	proveit.numbers.exponentiation.exp_real_closure_nat_power
70	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
71	theorem		⊢
	proveit.numbers.division.strong_div_from_denom_bound__all_pos
72	instantiation	246, 83, 82	⊢
	: , : , :
73	instantiation	246, 83, 84	,  ⊢
	: , : , :
74	instantiation	85, 207, 86, 121, 87, 88	,  ⊢
	: , : , :
75	instantiation	94, 102, 89	,  ⊢
	:
76	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
77	instantiation	90, 119, 136, 189, 91, 92*	,  ⊢
	: , : , :
78	axiom		⊢
	proveit.logic.equality.equals_reflexivity
79	instantiation	94, 119, 93	,  ⊢
	:
80	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
81	instantiation	94, 121, 95	,  ⊢
	:
82	instantiation	246, 96, 200	⊢
	: , : , :
83	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
84	instantiation	246, 96, 97	,  ⊢
	: , : , :
85	theorem		⊢
	proveit.numbers.exponentiation.exp_pos_less
86	instantiation	134, 135, 127	,  ⊢
	: , :
87	instantiation	98, 125, 99	,  ⊢
	: , :
88	instantiation	100, 101	⊢
	:
89	instantiation	120, 102, 189, 103	,  ⊢
	: , :
90	theorem		⊢
	proveit.numbers.addition.weak_bound_via_right_term_bound
91	instantiation	104, 127, 189, 153, 132, 105, 130*, 106*	⊢
	: , : , :
92	instantiation	214, 107	,  ⊢
	:
93	instantiation	108, 150, 109, 131	,  ⊢
	: , :
94	theorem		⊢
	proveit.numbers.exponentiation.sqrd_pos_closure
95	instantiation	110, 111	,  ⊢
	: , :
96	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
97	instantiation	112, 113, 248	,  ⊢
	: , :
98	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
99	instantiation	114, 135, 127, 136, 115	,  ⊢
	: , : , :
100	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
101	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
102	instantiation	134, 119, 136	,  ⊢
	: , :
103	instantiation	116, 117	,  ⊢
	: , :
104	theorem		⊢
	proveit.numbers.multiplication.reversed_weak_bound_via_right_factor_bound
105	instantiation	139, 128	⊢
	: , :
106	instantiation	118, 162	⊢
	:
107	instantiation	246, 226, 119	,  ⊢
	: , : , :
108	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq_int
109	theorem		⊢
	proveit.numbers.number_sets.integers.zero_is_int
110	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
111	instantiation	120, 189, 121, 122	,  ⊢
	: , :
112	theorem		⊢
	proveit.numbers.exponentiation.exp_natpos_closure
113	instantiation	123, 124, 125	,  ⊢
	:
114	theorem		⊢
	proveit.numbers.addition.strong_bound_via_right_term_bound
115	instantiation	126, 127, 153, 227, 155, 128, 129*, 130*	⊢
	: , : , :
116	theorem		⊢
	proveit.numbers.addition.subtraction.neg_difference
117	instantiation	222, 131, 132	,  ⊢
	: , : , :
118	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
119	instantiation	246, 234, 133	,  ⊢
	: , : , :
120	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq
121	instantiation	134, 135, 136	,  ⊢
	: , :
122	instantiation	158, 137	,  ⊢
	: , :
123	theorem		⊢
	proveit.numbers.number_sets.integers.nonneg_int_is_natural
124	instantiation	236, 170, 138	,  ⊢
	: , :
125	instantiation	139, 140	,  ⊢
	: , :
126	theorem		⊢
	proveit.numbers.multiplication.reversed_strong_bound_via_right_factor_bound
127	instantiation	152, 227	⊢
	:
128	instantiation	166, 157	⊢
	:
129	instantiation	141, 216, 142*	⊢
	: , :
130	instantiation	143, 144, 145	⊢
	: , : , :
131	instantiation	146, 147, 148	,  ⊢
	: , : , :
132	instantiation	149, 189, 227, 173	⊢
	: , : , :
133	instantiation	246, 239, 150	,  ⊢
	: , : , :
134	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
135	instantiation	246, 234, 151	,  ⊢
	: , : , :
136	instantiation	152, 153	⊢
	:
137	instantiation	154, 155, 159	,  ⊢
	: , : , :
138	instantiation	246, 156, 157	⊢
	: , : , :
139	theorem		⊢
	proveit.numbers.ordering.relax_less
140	instantiation	158, 159	,  ⊢
	: , :
141	theorem		⊢
	proveit.numbers.multiplication.mult_neg_left
142	instantiation	160, 216	⊢
	:
143	axiom		⊢
	proveit.logic.equality.equals_transitivity
144	instantiation	161, 245, 248, 178, 180, 179, 162, 181, 182	⊢
	: , : , : , : , : , :
145	instantiation	163, 178, 248, 179, 180, 216, 181, 182, 164*	⊢
	: , : , : , : , :
146	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less
147	instantiation	165, 185, 186, 169	,  ⊢
	: , : , :
148	instantiation	166, 167	⊢
	:
149	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_co_lower_bound
150	instantiation	246, 168, 169	,  ⊢
	: , : , :
151	instantiation	246, 239, 170	,  ⊢
	: , : , :
152	theorem		⊢
	proveit.numbers.negation.real_closure
153	instantiation	171, 189, 227, 173	⊢
	: , : , :
154	axiom		⊢
	proveit.numbers.ordering.transitivity_less_less
155	instantiation	172, 189, 227, 173	⊢
	: , : , :
156	theorem		⊢
	proveit.numbers.number_sets.integers.neg_int_within_int
157	instantiation	183, 200	⊢
	:
158	theorem		⊢
	proveit.numbers.addition.subtraction.pos_difference
159	instantiation	222, 174, 175	,  ⊢
	: , : , :
160	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
161	theorem		⊢
	proveit.numbers.multiplication.disassociation
162	instantiation	176, 216	⊢
	:
163	theorem		⊢
	proveit.numbers.multiplication.mult_neg_any
164	instantiation	177, 178, 248, 179, 180, 181, 182	⊢
	: , : , : , :
165	theorem		⊢
	proveit.numbers.number_sets.integers.interval_upper_bound
166	theorem		⊢
	proveit.numbers.number_sets.integers.negative_if_in_neg_int
167	instantiation	183, 184	⊢
	:
168	instantiation	232, 185, 186	⊢
	: , :
169	assumption		⊢
170	instantiation	246, 187, 192	,  ⊢
	: , : , :
171	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_co__is__real
172	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_co_upper_bound
173	theorem		⊢
	proveit.physics.quantum.QPE._scaled_delta_b_floor_in_interval
174	instantiation	188, 227, 189, 190, 212, 191*	⊢
	: , : , :
175	instantiation	230, 202, 237, 192	,  ⊢
	: , : , :
176	theorem		⊢
	proveit.numbers.negation.complex_closure
177	theorem		⊢
	proveit.numbers.multiplication.elim_one_any
178	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
179	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
180	instantiation	193	⊢
	: , :
181	instantiation	194, 195, 196	⊢
	: , :
182	instantiation	246, 226, 197	⊢
	: , : , :
183	theorem		⊢
	proveit.numbers.negation.int_neg_closure
184	instantiation	198, 199, 200	⊢
	: , :
185	instantiation	236, 201, 240	⊢
	: , :
186	instantiation	243, 202	⊢
	:
187	instantiation	232, 202, 237	⊢
	: , :
188	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
189	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
190	instantiation	246, 234, 203	⊢
	: , : , :
191	instantiation	204, 205, 206	⊢
	: , : , :
192	assumption		⊢
193	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
194	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
195	instantiation	246, 226, 207	⊢
	: , : , :
196	instantiation	246, 226, 208	⊢
	: , : , :
197	instantiation	209, 210	⊢
	:
198	theorem		⊢
	proveit.numbers.addition.add_nat_pos_closure_bin
199	instantiation	211, 213, 212	⊢
	:
200	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
201	instantiation	243, 237	⊢
	:
202	instantiation	236, 213, 240	⊢
	: , :
203	instantiation	246, 239, 213	⊢
	: , : , :
204	theorem		⊢
	proveit.logic.equality.sub_right_side_into
205	instantiation	214, 216	⊢
	:
206	instantiation	215, 216, 217	⊢
	: , :
207	instantiation	246, 234, 218	⊢
	: , : , :
208	instantiation	219, 220, 221	⊢
	: , : , :
209	theorem		⊢
	proveit.physics.quantum.QPE._delta_b_is_real
210	theorem		⊢
	proveit.physics.quantum.QPE._best_floor_is_int
211	theorem		⊢
	proveit.numbers.number_sets.integers.pos_int_is_natural_pos
212	instantiation	222, 223, 224	⊢
	: , : , :
213	instantiation	246, 225, 231	⊢
	: , : , :
214	theorem		⊢
	proveit.numbers.addition.elim_zero_right
215	theorem		⊢
	proveit.numbers.addition.commutation
216	instantiation	246, 226, 227	⊢
	: , : , :
217	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.zero_is_complex
218	instantiation	246, 239, 244	⊢
	: , : , :
219	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
220	instantiation	228, 229	⊢
	: , :
221	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
222	theorem		⊢
	proveit.numbers.ordering.transitivity_less_less_eq
223	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
224	instantiation	230, 240, 233, 231	⊢
	: , : , :
225	instantiation	232, 240, 233	⊢
	: , :
226	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
227	instantiation	246, 234, 235	⊢
	: , : , :
228	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
229	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
230	theorem		⊢
	proveit.numbers.number_sets.integers.interval_lower_bound
231	assumption		⊢
232	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
233	instantiation	236, 237, 238	⊢
	: , :
234	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
235	instantiation	246, 239, 240	⊢
	: , : , :
236	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
237	instantiation	246, 241, 242	⊢
	: , : , :
238	instantiation	243, 244	⊢
	:
239	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
240	instantiation	246, 247, 245	⊢
	: , : , :
241	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
242	theorem		⊢
	proveit.physics.quantum.QPE._two_pow_t_minus_one_is_nat_pos
243	theorem		⊢
	proveit.numbers.negation.int_closure
244	instantiation	246, 247, 248	⊢
	: , : , :
245	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
246	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
247	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
248	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
*equality replacement requirements