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	reference	238	⊢
2	instantiation	4, 5, 6	⊢
	: , : , :
3	instantiation	7, 236, 271, 172, 8*, 9*	⊢
	: , : , : , : , :
4	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
5	instantiation	12, 13, 10, 14, 11, 17	⊢
	: , : , :
6	instantiation	12, 13, 14, 15, 16, 17	⊢
	: , : , :
7	theorem		⊢
	proveit.numbers.summation.index_shift
8	instantiation	177, 18, 19	⊢
	: , : , :
9	instantiation	177, 20, 21	,  ⊢
	: , : , :
10	instantiation	168, 23, 27	⊢
	: , :
11	instantiation	22, 27, 23, 26, 24	⊢
	: , : , :
12	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
13	instantiation	280, 268, 25	⊢
	: , : , :
14	instantiation	168, 26, 27	⊢
	: , :
15	instantiation	168, 26, 28	⊢
	: , :
16	instantiation	123, 26, 27, 28, 29	⊢
	: , : , :
17	instantiation	173, 30	⊢
	: , :
18	instantiation	56, 212, 282, 279, 213, 31, 32, 250, 196	⊢
	: , : , : , : , : , :
19	instantiation	58, 250, 32, 60	⊢
	: , : , :
20	instantiation	71, 33	⊢
	: , : , :
21	instantiation	177, 34, 35	,  ⊢
	: , : , :
22	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
23	modus ponens	36, 37	⊢
24	modus ponens	38, 39	⊢
25	instantiation	280, 40, 50	⊢
	: , : , :
26	modus ponens	41, 42	⊢
27	modus ponens	43, 44	⊢
28	modus ponens	45, 46	⊢
29	modus ponens	47, 48	⊢
30	instantiation	49, 50	⊢
	:
31	instantiation	227	⊢
	: , :
32	instantiation	280, 260, 224	⊢
	: , : , :
33	instantiation	51, 134, 52, 53, 54, 55	⊢
	: , : , :
34	instantiation	56, 212, 282, 279, 213, 57, 59, 250, 196	,  ⊢
	: , : , : , : , : , :
35	instantiation	58, 250, 59, 60	,  ⊢
	: , : , :
36	instantiation	65	⊢
	: , : , :
37	generalization	61	⊢
38	instantiation	67	⊢
	: , : , :
39	generalization	62	⊢
40	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
41	instantiation	65	⊢
	: , : , :
42	generalization	63	⊢
43	instantiation	65	⊢
	: , : , :
44	generalization	64	⊢
45	instantiation	65	⊢
	: , : , :
46	generalization	66	⊢
47	instantiation	67	⊢
	: , : , :
48	generalization	68	⊢
49	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
50	instantiation	69, 113, 70	⊢
	: , :
51	theorem		⊢
	proveit.core_expr_types.tuples.tuple_eq_via_elem_eq
52	instantiation	227	⊢
	: , :
53	instantiation	227	⊢
	: , :
54	instantiation	71, 72	⊢
	: , : , :
55	instantiation	109	⊢
	:
56	theorem		⊢
	proveit.numbers.addition.disassociation
57	instantiation	227	⊢
	: , :
58	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_23
59	instantiation	280, 260, 73	,  ⊢
	: , : , :
60	instantiation	109	⊢
	:
61	instantiation	83, 261, 74, 75	,  ⊢
	: , :
62	instantiation	76, 77, 110, 106, 78	,  ⊢
	: , : , :
63	instantiation	83, 261, 79, 80	,  ⊢
	: , :
64	instantiation	83, 261, 81, 82	,  ⊢
	: , :
65	theorem		⊢
	proveit.numbers.summation.summation_real_closure
66	instantiation	83, 261, 84, 85	,  ⊢
	: , :
67	theorem		⊢
	proveit.numbers.summation.weak_summation_from_summands_bound
68	instantiation	173, 86	,  ⊢
	: , :
69	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
70	instantiation	280, 129, 87	⊢
	: , : , :
71	axiom		⊢
	proveit.logic.equality.substitution
72	instantiation	88, 250	⊢
	:
73	instantiation	280, 268, 89	,  ⊢
	: , : , :
74	instantiation	99, 136, 282	,  ⊢
	: , :
75	instantiation	97, 90	,  ⊢
	:
76	theorem		⊢
	proveit.numbers.division.weak_div_from_denom_bound__all_pos
77	instantiation	280, 91, 92	⊢
	: , : , :
78	instantiation	93, 241, 136, 153, 94, 95	,  ⊢
	: , : , :
79	instantiation	99, 153, 282	,  ⊢
	: , :
80	instantiation	97, 96	,  ⊢
	:
81	instantiation	99, 155, 282	,  ⊢
	: , :
82	instantiation	97, 98	,  ⊢
	:
83	theorem		⊢
	proveit.numbers.division.div_real_closure
84	instantiation	99, 117, 282	,  ⊢
	: , :
85	instantiation	100, 130	,  ⊢
	:
86	instantiation	101, 102, 103, 112, 104	,  ⊢
	: , : , :
87	theorem		⊢
	proveit.numbers.numerals.decimals.posnat4
88	theorem		⊢
	proveit.numbers.negation.double_negation
89	instantiation	280, 273, 105	,  ⊢
	: , : , :
90	instantiation	280, 111, 106	,  ⊢
	: , : , :
91	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
92	instantiation	280, 107, 279	⊢
	: , : , :
93	theorem		⊢
	proveit.numbers.exponentiation.exp_even_neg_base_lesseq
94	instantiation	131, 108, 165	,  ⊢
	: , :
95	instantiation	109	⊢
	:
96	instantiation	280, 111, 110	,  ⊢
	: , : , :
97	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
98	instantiation	280, 111, 112	,  ⊢
	: , : , :
99	theorem		⊢
	proveit.numbers.exponentiation.exp_real_closure_nat_power
100	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
101	theorem		⊢
	proveit.numbers.division.strong_div_from_denom_bound__all_pos
102	instantiation	280, 114, 113	⊢
	: , : , :
103	instantiation	280, 114, 115	,  ⊢
	: , : , :
104	instantiation	116, 241, 117, 155, 118, 119	,  ⊢
	: , : , :
105	instantiation	280, 120, 121	,  ⊢
	: , : , :
106	instantiation	127, 136, 122	,  ⊢
	:
107	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
108	instantiation	123, 153, 170, 223, 124, 125*	,  ⊢
	: , : , :
109	axiom		⊢
	proveit.logic.equality.equals_reflexivity
110	instantiation	127, 153, 126	,  ⊢
	:
111	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
112	instantiation	127, 155, 128	,  ⊢
	:
113	instantiation	280, 129, 234	⊢
	: , : , :
114	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
115	instantiation	280, 129, 130	,  ⊢
	: , : , :
116	theorem		⊢
	proveit.numbers.exponentiation.exp_pos_less
117	instantiation	168, 169, 161	,  ⊢
	: , :
118	instantiation	131, 159, 132	,  ⊢
	: , :
119	instantiation	133, 134	⊢
	:
120	instantiation	266, 247, 135	⊢
	: , :
121	assumption		⊢
122	instantiation	154, 136, 223, 137	,  ⊢
	: , :
123	theorem		⊢
	proveit.numbers.addition.weak_bound_via_right_term_bound
124	instantiation	138, 161, 223, 187, 166, 139, 164*, 140*	⊢
	: , : , :
125	instantiation	248, 141	,  ⊢
	:
126	instantiation	142, 184, 143, 165	,  ⊢
	: , :
127	theorem		⊢
	proveit.numbers.exponentiation.sqrd_pos_closure
128	instantiation	144, 145	,  ⊢
	: , :
129	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
130	instantiation	146, 147, 282	,  ⊢
	: , :
131	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
132	instantiation	148, 169, 161, 170, 149	,  ⊢
	: , : , :
133	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
134	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
135	instantiation	270, 271, 172	⊢
	: , :
136	instantiation	168, 153, 170	,  ⊢
	: , :
137	instantiation	150, 151	,  ⊢
	: , :
138	theorem		⊢
	proveit.numbers.multiplication.reversed_weak_bound_via_right_factor_bound
139	instantiation	173, 162	⊢
	: , :
140	instantiation	152, 196	⊢
	:
141	instantiation	280, 260, 153	,  ⊢
	: , : , :
142	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq_int
143	theorem		⊢
	proveit.numbers.number_sets.integers.zero_is_int
144	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
145	instantiation	154, 223, 155, 156	,  ⊢
	: , :
146	theorem		⊢
	proveit.numbers.exponentiation.exp_natpos_closure
147	instantiation	157, 158, 159	,  ⊢
	:
148	theorem		⊢
	proveit.numbers.addition.strong_bound_via_right_term_bound
149	instantiation	160, 161, 187, 261, 189, 162, 163*, 164*	⊢
	: , : , :
150	theorem		⊢
	proveit.numbers.addition.subtraction.neg_difference
151	instantiation	256, 165, 166	,  ⊢
	: , : , :
152	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
153	instantiation	280, 268, 167	,  ⊢
	: , : , :
154	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq
155	instantiation	168, 169, 170	,  ⊢
	: , :
156	instantiation	192, 171	,  ⊢
	: , :
157	theorem		⊢
	proveit.numbers.number_sets.integers.nonneg_int_is_natural
158	instantiation	270, 204, 172	,  ⊢
	: , :
159	instantiation	173, 174	,  ⊢
	: , :
160	theorem		⊢
	proveit.numbers.multiplication.reversed_strong_bound_via_right_factor_bound
161	instantiation	186, 261	⊢
	:
162	instantiation	200, 191	⊢
	:
163	instantiation	175, 250, 176*	⊢
	: , :
164	instantiation	177, 178, 179	⊢
	: , : , :
165	instantiation	180, 181, 182	,  ⊢
	: , : , :
166	instantiation	183, 223, 261, 207	⊢
	: , : , :
167	instantiation	280, 273, 184	,  ⊢
	: , : , :
168	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
169	instantiation	280, 268, 185	,  ⊢
	: , : , :
170	instantiation	186, 187	⊢
	:
171	instantiation	188, 189, 193	,  ⊢
	: , : , :
172	instantiation	280, 190, 191	⊢
	: , : , :
173	theorem		⊢
	proveit.numbers.ordering.relax_less
174	instantiation	192, 193	,  ⊢
	: , :
175	theorem		⊢
	proveit.numbers.multiplication.mult_neg_left
176	instantiation	194, 250	⊢
	:
177	axiom		⊢
	proveit.logic.equality.equals_transitivity
178	instantiation	195, 279, 282, 212, 214, 213, 196, 215, 216	⊢
	: , : , : , : , : , :
179	instantiation	197, 212, 282, 213, 214, 250, 215, 216, 198*	⊢
	: , : , : , : , :
180	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less
181	instantiation	199, 219, 220, 203	,  ⊢
	: , : , :
182	instantiation	200, 201	⊢
	:
183	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_co_lower_bound
184	instantiation	280, 202, 203	,  ⊢
	: , : , :
185	instantiation	280, 273, 204	,  ⊢
	: , : , :
186	theorem		⊢
	proveit.numbers.negation.real_closure
187	instantiation	205, 223, 261, 207	⊢
	: , : , :
188	axiom		⊢
	proveit.numbers.ordering.transitivity_less_less
189	instantiation	206, 223, 261, 207	⊢
	: , : , :
190	theorem		⊢
	proveit.numbers.number_sets.integers.neg_int_within_int
191	instantiation	217, 234	⊢
	:
192	theorem		⊢
	proveit.numbers.addition.subtraction.pos_difference
193	instantiation	256, 208, 209	,  ⊢
	: , : , :
194	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
195	theorem		⊢
	proveit.numbers.multiplication.disassociation
196	instantiation	210, 250	⊢
	:
197	theorem		⊢
	proveit.numbers.multiplication.mult_neg_any
198	instantiation	211, 212, 282, 213, 214, 215, 216	⊢
	: , : , : , :
199	theorem		⊢
	proveit.numbers.number_sets.integers.interval_upper_bound
200	theorem		⊢
	proveit.numbers.number_sets.integers.negative_if_in_neg_int
201	instantiation	217, 218	⊢
	:
202	instantiation	266, 219, 220	⊢
	: , :
203	assumption		⊢
204	instantiation	280, 221, 226	,  ⊢
	: , : , :
205	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_co__is__real
206	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_co_upper_bound
207	theorem		⊢
	proveit.physics.quantum.QPE._scaled_delta_b_floor_in_interval
208	instantiation	222, 261, 223, 224, 246, 225*	⊢
	: , : , :
209	instantiation	264, 236, 271, 226	,  ⊢
	: , : , :
210	theorem		⊢
	proveit.numbers.negation.complex_closure
211	theorem		⊢
	proveit.numbers.multiplication.elim_one_any
212	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
213	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
214	instantiation	227	⊢
	: , :
215	instantiation	228, 229, 230	⊢
	: , :
216	instantiation	280, 260, 231	⊢
	: , : , :
217	theorem		⊢
	proveit.numbers.negation.int_neg_closure
218	instantiation	232, 233, 234	⊢
	: , :
219	instantiation	270, 235, 274	⊢
	: , :
220	instantiation	277, 236	⊢
	:
221	instantiation	266, 236, 271	⊢
	: , :
222	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
223	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
224	instantiation	280, 268, 237	⊢
	: , : , :
225	instantiation	238, 239, 240	⊢
	: , : , :
226	assumption		⊢
227	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
228	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
229	instantiation	280, 260, 241	⊢
	: , : , :
230	instantiation	280, 260, 242	⊢
	: , : , :
231	instantiation	243, 244	⊢
	:
232	theorem		⊢
	proveit.numbers.addition.add_nat_pos_closure_bin
233	instantiation	245, 247, 246	⊢
	:
234	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
235	instantiation	277, 271	⊢
	:
236	instantiation	270, 247, 274	⊢
	: , :
237	instantiation	280, 273, 247	⊢
	: , : , :
238	theorem		⊢
	proveit.logic.equality.sub_right_side_into
239	instantiation	248, 250	⊢
	:
240	instantiation	249, 250, 251	⊢
	: , :
241	instantiation	280, 268, 252	⊢
	: , : , :
242	instantiation	253, 254, 255	⊢
	: , : , :
243	theorem		⊢
	proveit.physics.quantum.QPE._delta_b_is_real
244	theorem		⊢
	proveit.physics.quantum.QPE._best_floor_is_int
245	theorem		⊢
	proveit.numbers.number_sets.integers.pos_int_is_natural_pos
246	instantiation	256, 257, 258	⊢
	: , : , :
247	instantiation	280, 259, 265	⊢
	: , : , :
248	theorem		⊢
	proveit.numbers.addition.elim_zero_right
249	theorem		⊢
	proveit.numbers.addition.commutation
250	instantiation	280, 260, 261	⊢
	: , : , :
251	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.zero_is_complex
252	instantiation	280, 273, 278	⊢
	: , : , :
253	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
254	instantiation	262, 263	⊢
	: , :
255	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
256	theorem		⊢
	proveit.numbers.ordering.transitivity_less_less_eq
257	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
258	instantiation	264, 274, 267, 265	⊢
	: , : , :
259	instantiation	266, 274, 267	⊢
	: , :
260	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
261	instantiation	280, 268, 269	⊢
	: , : , :
262	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
263	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
264	theorem		⊢
	proveit.numbers.number_sets.integers.interval_lower_bound
265	assumption		⊢
266	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
267	instantiation	270, 271, 272	⊢
	: , :
268	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
269	instantiation	280, 273, 274	⊢
	: , : , :
270	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
271	instantiation	280, 275, 276	⊢
	: , : , :
272	instantiation	277, 278	⊢
	:
273	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
274	instantiation	280, 281, 279	⊢
	: , : , :
275	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
276	theorem		⊢
	proveit.physics.quantum.QPE._two_pow_t_minus_one_is_nat_pos
277	theorem		⊢
	proveit.numbers.negation.int_closure
278	instantiation	280, 281, 282	⊢
	: , : , :
279	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
280	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
281	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
282	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
*equality replacement requirements