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, 6, 7*	,  ⊢
	: , : , :
1	theorem		⊢
	proveit.numbers.division.weak_div_from_numer_bound__pos_denom
2	reference	232	⊢
3	instantiation	231, 232, 31	⊢
	: , :
4	reference	77	⊢
5	instantiation	154, 8, 9	,  ⊢
	: , : , :
6	instantiation	10, 114	⊢
	:
7	instantiation	203, 11, 12	⊢
	: , : , :
8	instantiation	32, 13, 14	,  ⊢
	: , : , :
9	instantiation	140, 15, 16	⊢
	: , : , :
10	theorem		⊢
	proveit.numbers.number_sets.real_numbers.positive_if_in_real_pos
11	instantiation	161, 79, 163, 17, 117*	⊢
	: , : , :
12	instantiation	180, 17	⊢
	:
13	instantiation	18, 92, 19, 243, 20, 21*	⊢
	: , : , :
14	instantiation	22, 243, 92, 23, 24, 25*	,  ⊢
	: , : , :
15	instantiation	26, 232, 43, 243, 117*	⊢
	: , : , :
16	instantiation	27, 265, 216, 218, 150, 28, 29, 30*	⊢
	: , : , : , : , : , :
17	instantiation	263, 236, 31	⊢
	: , : , :
18	theorem		⊢
	proveit.numbers.addition.weak_bound_via_right_term_bound
19	instantiation	190, 76, 75	⊢
	: , :
20	instantiation	32, 33, 34	⊢
	: , : , :
21	instantiation	203, 35, 36	⊢
	: , : , :
22	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
23	instantiation	190, 77, 235	⊢
	: , :
24	instantiation	140, 37, 38	⊢
	: , : , :
25	instantiation	203, 39, 40	⊢
	: , : , :
26	theorem		⊢
	proveit.numbers.modular.mod_abs_scaled
27	theorem		⊢
	proveit.numbers.multiplication.distribute_through_subtract
28	instantiation	263, 236, 63	⊢
	: , : , :
29	instantiation	263, 236, 233	⊢
	: , : , :
30	instantiation	140, 41, 42	⊢
	: , : , :
31	instantiation	108, 43, 243, 44	⊢
	: , :
32	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
33	instantiation	140, 45, 46, 47*	⊢
	: , : , :
34	instantiation	154, 48, 49	⊢
	: , : , :
35	instantiation	215, 265, 260, 216, 50, 218, 52, 53, 51	⊢
	: , : , : , : , : , :
36	instantiation	220, 52, 53, 54	⊢
	: , : , :
37	assumption		⊢
38	axiom		⊢
	proveit.physics.quantum.QPE._best_floor_def
39	instantiation	215, 216, 260, 265, 218, 55, 57, 219, 221	⊢
	: , : , : , : , : , :
40	instantiation	56, 221, 57, 223	⊢
	: , : , :
41	instantiation	140, 58, 59	⊢
	: , : , :
42	instantiation	99, 60, 61, 62	⊢
	: , : , : , :
43	instantiation	190, 63, 64	⊢
	: , :
44	instantiation	65, 227	⊢
	:
45	instantiation	66, 93, 109, 114, 67*	⊢
	: , : , :
46	instantiation	73, 123, 158	⊢
	: , :
47	instantiation	68, 69, 114, 70, 71*	⊢
	: , :
48	instantiation	87, 72	⊢
	: , :
49	instantiation	73, 74, 157	⊢
	: , :
50	instantiation	234	⊢
	: , :
51	instantiation	263, 236, 75	⊢
	: , : , :
52	instantiation	263, 236, 92	⊢
	: , : , :
53	instantiation	263, 236, 76	⊢
	: , : , :
54	instantiation	238	⊢
	:
55	instantiation	234	⊢
	: , :
56	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
57	instantiation	263, 236, 77	⊢
	: , : , :
58	instantiation	78, 221, 152, 79, 163	⊢
	: , : , : , : , :
59	instantiation	203, 80, 81	⊢
	: , : , :
60	instantiation	159, 82	⊢
	: , : , :
61	instantiation	159, 83	⊢
	: , : , :
62	instantiation	149, 152	⊢
	:
63	instantiation	84, 167, 232, 110	⊢
	: , :
64	instantiation	153, 233	⊢
	:
65	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
66	theorem		⊢
	proveit.numbers.modular.mod_abs_of_difference_bound
67	instantiation	203, 85, 86	⊢
	: , : , :
68	theorem		⊢
	proveit.numbers.modular.mod_abs_x_reduce_to_abs_x
69	instantiation	190, 170, 139	⊢
	: , :
70	instantiation	87, 88	⊢
	: , :
71	instantiation	89, 90, 91*	⊢
	:
72	instantiation	125, 242, 243, 186	⊢
	: , : , :
73	theorem		⊢
	proveit.numbers.addition.commutation
74	instantiation	263, 236, 130	⊢
	: , : , :
75	instantiation	153, 92	⊢
	:
76	instantiation	108, 93, 232, 110	⊢
	: , :
77	instantiation	263, 249, 94	⊢
	: , : , :
78	theorem		⊢
	proveit.numbers.division.mult_frac_cancel_numer_left
79	instantiation	263, 177, 95	⊢
	: , : , :
80	instantiation	159, 96	⊢
	: , : , :
81	instantiation	159, 97	⊢
	: , : , :
82	instantiation	180, 221	⊢
	:
83	instantiation	180, 152	⊢
	:
84	theorem		⊢
	proveit.numbers.division.div_real_closure
85	instantiation	159, 98	⊢
	: , : , :
86	instantiation	99, 100, 101, 102	⊢
	: , : , : , :
87	theorem		⊢
	proveit.numbers.ordering.relax_less
88	instantiation	103, 104, 105	⊢
	: , : , :
89	theorem		⊢
	proveit.numbers.absolute_value.abs_neg_elim
90	instantiation	106, 107	⊢
	: , :
91	instantiation	118, 158, 157	⊢
	: , :
92	instantiation	108, 109, 232, 110	⊢
	: , :
93	instantiation	190, 167, 139	⊢
	: , :
94	instantiation	263, 111, 112	⊢
	: , : , :
95	instantiation	263, 113, 114	⊢
	: , : , :
96	instantiation	203, 115, 116	⊢
	: , : , :
97	instantiation	159, 117	⊢
	: , : , :
98	instantiation	118, 152, 158	⊢
	: , :
99	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
100	instantiation	215, 216, 260, 265, 218, 120, 152, 123, 119	⊢
	: , : , : , : , : , :
101	instantiation	215, 260, 216, 120, 121, 218, 152, 123, 138, 158	⊢
	: , : , : , : , : , :
102	instantiation	122, 216, 265, 218, 152, 123, 158, 124	⊢
	: , : , : , : , : , : , : , :
103	theorem		⊢
	proveit.numbers.ordering.transitivity_less_less_eq
104	instantiation	125, 242, 243, 126	⊢
	: , : , :
105	instantiation	154, 127, 128	⊢
	: , : , :
106	theorem		⊢
	proveit.numbers.addition.subtraction.nonpos_difference
107	instantiation	129, 214	⊢
	:
108	theorem		⊢
	proveit.numbers.modular.mod_abs_real_closure
109	instantiation	190, 167, 130	⊢
	: , :
110	instantiation	131, 195, 132	⊢
	: , :
111	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_nonzero_within_rational
112	instantiation	133, 195, 134	⊢
	: , :
113	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
114	instantiation	135, 172, 237	⊢
	: , :
115	instantiation	159, 136	⊢
	: , : , :
116	instantiation	180, 150	⊢
	:
117	instantiation	179, 150	⊢
	:
118	theorem		⊢
	proveit.numbers.negation.distribute_neg_through_subtract
119	instantiation	137, 138, 158	⊢
	: , :
120	instantiation	234	⊢
	: , :
121	instantiation	234	⊢
	: , :
122	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_general
123	instantiation	263, 236, 139	⊢
	: , : , :
124	instantiation	238	⊢
	:
125	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_co_upper_bound
126	instantiation	140, 141, 142	⊢
	: , : , :
127	instantiation	143, 210	⊢
	:
128	instantiation	203, 144, 145	⊢
	: , : , :
129	theorem		⊢
	proveit.numbers.rounding.floor_x_le_x
130	instantiation	153, 170	⊢
	:
131	theorem		⊢
	proveit.numbers.exponentiation.exp_rational_non_zero__not_zero
132	instantiation	263, 208, 146	⊢
	: , : , :
133	theorem		⊢
	proveit.numbers.exponentiation.exp_rational_nonzero_closure
134	instantiation	239, 147, 148	⊢
	: , :
135	theorem		⊢
	proveit.numbers.exponentiation.exp_real_pos_closure
136	instantiation	149, 150	⊢
	:
137	theorem		⊢
	proveit.numbers.addition.add_complex_closure_bin
138	instantiation	151, 152	⊢
	:
139	instantiation	153, 214	⊢
	:
140	theorem		⊢
	proveit.logic.equality.sub_right_side_into
141	instantiation	154, 186, 155	⊢
	: , : , :
142	instantiation	156, 157, 158	⊢
	: , :
143	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_lower_bound
144	instantiation	159, 160	⊢
	: , : , :
145	instantiation	161, 162, 163, 181, 164*, 165*	⊢
	: , : , :
146	instantiation	263, 226, 262	⊢
	: , : , :
147	instantiation	263, 182, 262	⊢
	: , : , :
148	instantiation	258, 166	⊢
	:
149	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
150	instantiation	263, 236, 232	⊢
	: , : , :
151	theorem		⊢
	proveit.numbers.negation.complex_closure
152	instantiation	263, 236, 167	⊢
	: , : , :
153	theorem		⊢
	proveit.numbers.negation.real_closure
154	theorem		⊢
	proveit.logic.equality.sub_left_side_into
155	instantiation	168, 169	⊢
	:
156	theorem		⊢
	proveit.numbers.absolute_value.abs_diff_reversal
157	instantiation	263, 236, 214	⊢
	: , : , :
158	instantiation	263, 236, 170	⊢
	: , : , :
159	axiom		⊢
	proveit.logic.equality.substitution
160	instantiation	171, 172, 237, 243, 173, 174, 175*	⊢
	: , : , : , :
161	theorem		⊢
	proveit.numbers.division.frac_cancel_left
162	instantiation	263, 177, 176	⊢
	: , : , :
163	instantiation	263, 177, 178	⊢
	: , : , :
164	instantiation	179, 194	⊢
	:
165	instantiation	180, 181	⊢
	:
166	instantiation	263, 182, 183	⊢
	: , : , :
167	instantiation	263, 249, 184	⊢
	: , : , :
168	theorem		⊢
	proveit.numbers.absolute_value.abs_non_neg_elim
169	instantiation	185, 242, 243, 186	⊢
	: , : , :
170	instantiation	263, 249, 187	⊢
	: , : , :
171	theorem		⊢
	proveit.numbers.exponentiation.exp_factored_real
172	instantiation	263, 188, 189	⊢
	: , : , :
173	instantiation	190, 237, 235	⊢
	: , :
174	instantiation	191, 192	⊢
	: , :
175	instantiation	193, 194	⊢
	:
176	instantiation	263, 196, 195	⊢
	: , : , :
177	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
178	instantiation	263, 196, 197	⊢
	: , : , :
179	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
180	theorem		⊢
	proveit.numbers.division.frac_one_denom
181	instantiation	263, 236, 198	⊢
	: , : , :
182	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
183	axiom		⊢
	proveit.physics.quantum.QPE._n_in_natural_pos
184	instantiation	263, 257, 199	⊢
	: , : , :
185	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_co_lower_bound
186	instantiation	200, 214	⊢
	:
187	instantiation	263, 257, 201	⊢
	: , : , :
188	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
189	instantiation	263, 202, 225	⊢
	: , : , :
190	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
191	theorem		⊢
	proveit.logic.equality.equals_reversal
192	instantiation	203, 204, 205	⊢
	: , : , :
193	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
194	instantiation	263, 236, 206	⊢
	: , : , :
195	instantiation	263, 208, 207	⊢
	: , : , :
196	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
197	instantiation	263, 208, 209	⊢
	: , : , :
198	instantiation	246, 247, 210	⊢
	: , : , :
199	instantiation	263, 211, 212	⊢
	: , : , :
200	theorem		⊢
	proveit.numbers.rounding.real_minus_floor_interval
201	instantiation	213, 214	⊢
	:
202	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
203	axiom		⊢
	proveit.logic.equality.equals_transitivity
204	instantiation	215, 265, 260, 216, 217, 218, 221, 222, 219	⊢
	: , : , : , : , : , :
205	instantiation	220, 221, 222, 223	⊢
	: , : , :
206	instantiation	263, 249, 224	⊢
	: , : , :
207	instantiation	263, 226, 225	⊢
	: , : , :
208	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
209	instantiation	263, 226, 227	⊢
	: , : , :
210	theorem		⊢
	proveit.physics.quantum.QPE._two_pow_t_minus_one_is_nat_pos
211	instantiation	228, 229, 230	⊢
	: , :
212	assumption		⊢
213	axiom		⊢
	proveit.numbers.rounding.floor_is_an_int
214	instantiation	231, 232, 233	⊢
	: , :
215	theorem		⊢
	proveit.numbers.addition.disassociation
216	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
217	instantiation	234	⊢
	: , :
218	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
219	instantiation	263, 236, 235	⊢
	: , : , :
220	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
221	instantiation	263, 236, 243	⊢
	: , : , :
222	instantiation	263, 236, 237	⊢
	: , : , :
223	instantiation	238	⊢
	:
224	instantiation	263, 257, 255	⊢
	: , : , :
225	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
226	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
227	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
228	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
229	theorem		⊢
	proveit.numbers.number_sets.integers.zero_is_int
230	instantiation	239, 248, 251	⊢
	: , :
231	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
232	instantiation	263, 249, 240	⊢
	: , : , :
233	instantiation	241, 242, 243, 244	⊢
	: , : , :
234	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
235	instantiation	263, 249, 245	⊢
	: , : , :
236	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
237	instantiation	246, 247, 262	⊢
	: , : , :
238	axiom		⊢
	proveit.logic.equality.equals_reflexivity
239	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
240	instantiation	263, 257, 248	⊢
	: , : , :
241	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_co__is__real
242	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
243	instantiation	263, 249, 250	⊢
	: , : , :
244	axiom		⊢
	proveit.physics.quantum.QPE._phase_in_interval
245	instantiation	263, 257, 251	⊢
	: , : , :
246	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
247	instantiation	252, 253	⊢
	: , :
248	instantiation	254, 255, 256	⊢
	: , :
249	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
250	instantiation	263, 257, 259	⊢
	: , : , :
251	instantiation	258, 259	⊢
	:
252	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
253	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
254	theorem		⊢
	proveit.numbers.exponentiation.exp_int_closure
255	instantiation	263, 264, 260	⊢
	: , : , :
256	instantiation	263, 261, 262	⊢
	: , : , :
257	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
258	theorem		⊢
	proveit.numbers.negation.int_closure
259	instantiation	263, 264, 265	⊢
	: , : , :
260	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
261	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
262	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
263	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
264	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
265	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements