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	211	⊢
2	instantiation	148, 4	⊢
	: , :
3	instantiation	148, 5	⊢
	: , :
4	instantiation	6, 301, 7, 8, 9, 10	⊢
	: , : , : , :
5	instantiation	11, 61, 12, 13, 14, 15, 16*	⊢
	: , : , : , :
6	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
7	instantiation	262	⊢
	: , :
8	instantiation	262	⊢
	: , :
9	instantiation	148, 17	⊢
	: , :
10	instantiation	218, 18	⊢
	: , : , :
11	theorem		⊢
	proveit.logic.equality.sub_in_right_operands_via_tuple
12	instantiation	114, 19, 20, 23	⊢
	: , : , : , :
13	instantiation	114, 21, 22, 23	⊢
	: , : , : , :
14	instantiation	24, 78, 25	⊢
	:
15	instantiation	26, 27, 134*	⊢
	: , : , :
16	instantiation	28, 92, 246, 81, 247, 67, 29, 154, 30, 31, 32, 33	⊢
	: , : , : , : , : , : , : , : , : , :
17	instantiation	218, 34	⊢
	: , : , :
18	instantiation	148, 52	⊢
	: , :
19	instantiation	225, 35, 36	⊢
	: , : , :
20	instantiation	223	⊢
	:
21	instantiation	37, 297, 38, 39, 40, 41, 81, 60*, 67*	⊢
	: , : , : , :
22	instantiation	148, 42	⊢
	: , :
23	instantiation	148, 43	⊢
	: , :
24	axiom		⊢
	proveit.physics.quantum.QPE._psi_t_def
25	instantiation	211, 44, 45	⊢
	: , : , :
26	theorem		⊢
	proveit.core_expr_types.tuples.partition_front
27	instantiation	46, 246, 77	⊢
	: , :
28	theorem		⊢
	proveit.linear_algebra.tensors.factor_scalar_from_tensor_prod
29	instantiation	114, 47, 205, 50	⊢
	: , : , : , :
30	instantiation	48, 307, 308, 154, 72	⊢
	: , : , :
31	instantiation	114, 49, 205, 50	⊢
	: , : , : , :
32	instantiation	225, 51, 52	⊢
	: , : , :
33	modus ponens	53, 54	⊢
34	instantiation	218, 55	⊢
	: , : , :
35	instantiation	80, 56	⊢
	: , : , :
36	instantiation	211, 57, 58	⊢
	: , : , :
37	theorem		⊢
	proveit.core_expr_types.tuples.general_len
38	instantiation	262	⊢
	: , :
39	instantiation	262	⊢
	: , :
40	instantiation	262	⊢
	: , :
41	instantiation	59, 315, 60	⊢
	: , : , :
42	instantiation	202, 233, 234	⊢
	: , :
43	instantiation	84, 61	⊢
	: , :
44	instantiation	218, 62	⊢
	: , : , :
45	instantiation	211, 63, 64	⊢
	: , : , :
46	theorem		⊢
	proveit.numbers.addition.add_nat_closure_bin
47	instantiation	225, 65, 67	⊢
	: , : , :
48	theorem		⊢
	proveit.logic.booleans.conjunction.redundant_conjunction_general
49	instantiation	225, 66, 67	⊢
	: , : , :
50	instantiation	148, 68	⊢
	: , :
51	instantiation	120, 154, 121, 69	⊢
	: , : , : , :
52	instantiation	218, 70	⊢
	: , : , :
53	instantiation	71, 307, 308, 72	⊢
	: , : , : , :
54	generalization	73	⊢
55	instantiation	211, 74, 75	⊢
	: , : , :
56	instantiation	100, 229, 76, 246, 77, 315	⊢
	: , :
57	instantiation	218, 134	⊢
	: , : , :
58	instantiation	145, 246, 301, 247, 216, 233, 234	⊢
	: , : , : , :
59	theorem		⊢
	proveit.logic.equality.sub_left_side_into
60	instantiation	171, 234, 139	⊢
	: , : , :
61	instantiation	316, 284, 78	⊢
	: , : , :
62	instantiation	133, 233, 234	⊢
	: , :
63	instantiation	169, 246, 301, 315, 247, 79, 203, 137, 234	⊢
	: , : , : , : , : , :
64	instantiation	138, 234, 203, 139	⊢
	: , : , :
65	instantiation	80, 81	⊢
	: , : , :
66	instantiation	80, 81	⊢
	: , : , :
67	instantiation	211, 82, 83	⊢
	: , : , :
68	instantiation	84, 272	⊢
	: , :
69	instantiation	153, 154, 85, 156	⊢
	: , : , : , :
70	instantiation	148, 86	⊢
	: , :
71	theorem		⊢
	proveit.logic.booleans.conjunction.conjunction_from_quantification
72	instantiation	184, 87, 88, 252, 89, 90*, 91*	⊢
	: , : , :
73	instantiation	153, 154, 92, 93	,  ⊢
	: , : , : , :
74	instantiation	211, 94, 95	⊢
	: , : , :
75	instantiation	211, 96, 97	⊢
	: , : , :
76	instantiation	251	⊢
	: , : , :
77	instantiation	316, 284, 98	⊢
	: , : , :
78	instantiation	99, 318, 295	⊢
	: , :
79	instantiation	262	⊢
	: , :
80	theorem		⊢
	proveit.core_expr_types.tuples.range_len
81	instantiation	100, 229, 101, 246, 102, 315	⊢
	: , :
82	instantiation	218, 103	⊢
	: , : , :
83	instantiation	114, 104, 105, 106	⊢
	: , : , : , :
84	theorem		⊢
	proveit.core_expr_types.tuples.range_from1_len
85	instantiation	273, 176, 107	⊢
	: , :
86	instantiation	218, 108	⊢
	: , : , :
87	instantiation	316, 299, 109	⊢
	: , : , :
88	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
89	instantiation	110, 111	⊢
	: , :
90	instantiation	211, 112, 113	⊢
	: , : , :
91	instantiation	114, 115, 139, 116	⊢
	: , : , : , :
92	instantiation	117, 234, 118, 119	⊢
	: , :
93	instantiation	120, 154, 121, 122	,  ⊢
	: , : , : , :
94	instantiation	218, 123	⊢
	: , : , :
95	instantiation	218, 124	⊢
	: , : , :
96	instantiation	169, 246, 301, 315, 247, 125, 159, 236, 126	⊢
	: , : , : , : , : , :
97	instantiation	127, 159, 236, 128	⊢
	: , : , :
98	instantiation	129, 130	⊢
	:
99	theorem		⊢
	proveit.numbers.addition.add_nat_pos_closure_bin
100	theorem		⊢
	proveit.numbers.addition.add_nat_closure
101	instantiation	251	⊢
	: , : , :
102	instantiation	131, 132	⊢
	:
103	instantiation	133, 203, 234, 134*	⊢
	: , :
104	instantiation	169, 315, 301, 135, 144, 233, 137, 234	⊢
	: , : , : , : , : , :
105	instantiation	145, 246, 229, 247, 136, 233, 137, 234	⊢
	: , : , : , :
106	instantiation	138, 234, 233, 139	⊢
	: , : , :
107	instantiation	225, 140, 141	⊢
	: , : , :
108	instantiation	218, 142	⊢
	: , : , :
109	instantiation	316, 302, 307	⊢
	: , : , :
110	theorem		⊢
	proveit.numbers.ordering.relax_less
111	instantiation	143, 318	⊢
	:
112	instantiation	169, 315, 301, 246, 170, 247, 144, 203, 234	⊢
	: , : , : , : , : , :
113	instantiation	145, 246, 301, 247, 170, 203, 234	⊢
	: , : , : , :
114	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
115	instantiation	211, 146, 147	⊢
	: , : , :
116	instantiation	148, 149	⊢
	: , :
117	theorem		⊢
	proveit.numbers.division.div_complex_closure
118	instantiation	150, 274	⊢
	:
119	instantiation	151, 270, 152	⊢
	: , :
120	theorem		⊢
	proveit.linear_algebra.addition.binary_closure
121	theorem		⊢
	proveit.physics.quantum.algebra.ket_zero_in_qubit_space
122	instantiation	153, 154, 155, 156	,  ⊢
	: , : , : , :
123	instantiation	180, 214, 274, 181, 157*	⊢
	: , :
124	instantiation	218, 158	⊢
	: , : , :
125	instantiation	262	⊢
	: , :
126	instantiation	286, 159	⊢
	:
127	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
128	instantiation	223	⊢
	:
129	theorem		⊢
	proveit.numbers.negation.nat_pos_closure
130	instantiation	160, 318	⊢
	:
131	theorem		⊢
	proveit.numbers.negation.nat_closure
132	instantiation	161, 307, 162	⊢
	:
133	theorem		⊢
	proveit.numbers.negation.distribute_neg_through_binary_sum
134	instantiation	163, 233	⊢
	:
135	instantiation	262	⊢
	: , :
136	instantiation	251	⊢
	: , : , :
137	instantiation	286, 234	⊢
	:
138	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
139	instantiation	223	⊢
	:
140	instantiation	259, 228, 164	⊢
	: , :
141	instantiation	211, 165, 166	⊢
	: , : , :
142	instantiation	167, 229, 315, 246, 168, 247, 274, 250, 260, 208, 249	⊢
	: , : , : , : , : , : , :
143	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
144	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.zero_is_complex
145	theorem		⊢
	proveit.numbers.addition.elim_zero_any
146	instantiation	169, 315, 301, 246, 170, 247, 233, 203, 234	⊢
	: , : , : , : , : , :
147	instantiation	171, 233, 234, 205	⊢
	: , : , :
148	theorem		⊢
	proveit.logic.equality.equals_reversal
149	instantiation	172, 234	⊢
	:
150	theorem		⊢
	proveit.numbers.exponentiation.sqrt_complex_closure
151	theorem		⊢
	proveit.numbers.exponentiation.exp_rational_non_zero__not_zero
152	instantiation	173, 174, 270	⊢
	: , :
153	theorem		⊢
	proveit.linear_algebra.scalar_multiplication.scalar_mult_closure
154	instantiation	175, 297	⊢
	:
155	instantiation	273, 176, 177	,  ⊢
	: , :
156	theorem		⊢
	proveit.physics.quantum.algebra.ket_one_in_qubit_space
157	instantiation	211, 178, 179	⊢
	: , : , :
158	instantiation	180, 233, 274, 181, 182*	⊢
	: , :
159	instantiation	316, 293, 183	⊢
	: , : , :
160	theorem		⊢
	proveit.numbers.negation.int_neg_closure
161	theorem		⊢
	proveit.numbers.number_sets.integers.nonpos_int_is_int_nonpos
162	instantiation	184, 222, 253, 252, 185, 186*, 187*	⊢
	: , : , :
163	theorem		⊢
	proveit.numbers.negation.double_negation
164	instantiation	225, 188, 189	⊢
	: , : , :
165	instantiation	245, 315, 229, 246, 190, 247, 228, 260, 249, 208	⊢
	: , : , : , : , : , :
166	instantiation	245, 246, 301, 229, 247, 230, 190, 274, 250, 260, 249, 208	⊢
	: , : , : , : , : , :
167	theorem		⊢
	proveit.numbers.multiplication.leftward_commutation
168	instantiation	251	⊢
	: , : , :
169	theorem		⊢
	proveit.numbers.addition.disassociation
170	instantiation	262	⊢
	: , :
171	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_12
172	theorem		⊢
	proveit.numbers.addition.elim_zero_left
173	theorem		⊢
	proveit.numbers.division.div_rational_nonzero_closure
174	instantiation	316, 282, 191	⊢
	: , : , :
175	theorem		⊢
	proveit.linear_algebra.complex_vec_set_is_vec_space
176	instantiation	316, 293, 192	⊢
	: , : , :
177	instantiation	225, 193, 194	,  ⊢
	: , : , :
178	instantiation	218, 219	⊢
	: , : , :
179	instantiation	211, 195, 196	⊢
	: , : , :
180	theorem		⊢
	proveit.numbers.division.div_as_mult
181	instantiation	197, 297	⊢
	:
182	instantiation	211, 198, 199	⊢
	: , : , :
183	instantiation	316, 299, 200	⊢
	: , : , :
184	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
185	instantiation	201, 318	⊢
	:
186	instantiation	202, 234, 203	⊢
	: , :
187	instantiation	204, 233, 205	⊢
	: , :
188	instantiation	259, 206, 208	⊢
	: , :
189	instantiation	245, 246, 301, 315, 247, 207, 260, 249, 208	⊢
	: , : , : , : , : , :
190	instantiation	251	⊢
	: , : , :
191	instantiation	316, 291, 295	⊢
	: , : , :
192	instantiation	316, 276, 209	⊢
	: , : , :
193	instantiation	259, 228, 210	,  ⊢
	: , :
194	instantiation	211, 212, 213	,  ⊢
	: , : , :
195	instantiation	220, 214, 236	⊢
	: , :
196	instantiation	215, 315, 301, 246, 216, 247, 236, 233, 234, 217*	⊢
	: , : , : , : , : , :
197	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
198	instantiation	218, 219	⊢
	: , : , :
199	instantiation	220, 233, 236	⊢
	: , :
200	instantiation	316, 280, 221	⊢
	: , : , :
201	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_lower_bound
202	theorem		⊢
	proveit.numbers.addition.commutation
203	instantiation	316, 293, 222	⊢
	: , : , :
204	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_basic
205	instantiation	223	⊢
	:
206	instantiation	259, 260, 249	⊢
	: , :
207	instantiation	262	⊢
	: , :
208	instantiation	316, 293, 224	⊢
	: , : , :
209	theorem		⊢
	proveit.numbers.number_sets.real_numbers.e_is_real_pos
210	instantiation	225, 226, 227	,  ⊢
	: , : , :
211	axiom		⊢
	proveit.logic.equality.equals_transitivity
212	instantiation	245, 315, 229, 246, 231, 247, 228, 260, 261, 249	,  ⊢
	: , : , : , : , : , :
213	instantiation	245, 246, 301, 229, 247, 230, 231, 274, 250, 260, 261, 249	,  ⊢
	: , : , : , : , : , :
214	instantiation	232, 233, 234	⊢
	: , :
215	theorem		⊢
	proveit.numbers.multiplication.distribute_through_sum
216	instantiation	262	⊢
	: , :
217	instantiation	235, 236	⊢
	:
218	axiom		⊢
	proveit.logic.equality.substitution
219	instantiation	237, 238, 295, 239*	⊢
	: , :
220	theorem		⊢
	proveit.numbers.multiplication.commutation
221	instantiation	240, 281, 241	⊢
	: , :
222	instantiation	316, 299, 242	⊢
	: , : , :
223	axiom		⊢
	proveit.logic.equality.equals_reflexivity
224	instantiation	316, 299, 243	⊢
	: , : , :
225	theorem		⊢
	proveit.logic.equality.sub_right_side_into
226	instantiation	259, 244, 249	,  ⊢
	: , :
227	instantiation	245, 246, 301, 315, 247, 248, 260, 261, 249	,  ⊢
	: , : , : , : , : , :
228	instantiation	259, 274, 250	⊢
	: , :
229	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
230	instantiation	262	⊢
	: , :
231	instantiation	251	⊢
	: , : , :
232	theorem		⊢
	proveit.numbers.addition.add_complex_closure_bin
233	instantiation	316, 293, 252	⊢
	: , : , :
234	instantiation	316, 293, 253	⊢
	: , : , :
235	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
236	instantiation	316, 293, 254	⊢
	: , : , :
237	theorem		⊢
	proveit.numbers.exponentiation.neg_power_as_div
238	instantiation	316, 255, 256	⊢
	: , : , :
239	instantiation	257, 274	⊢
	:
240	theorem		⊢
	proveit.numbers.multiplication.mult_rational_pos_closure_bin
241	instantiation	316, 296, 318	⊢
	: , : , :
242	instantiation	316, 302, 310	⊢
	: , : , :
243	instantiation	316, 302, 258	⊢
	: , : , :
244	instantiation	259, 260, 261	,  ⊢
	: , :
245	theorem		⊢
	proveit.numbers.multiplication.disassociation
246	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
247	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
248	instantiation	262	⊢
	: , :
249	instantiation	316, 293, 263	⊢
	: , : , :
250	instantiation	316, 293, 264	⊢
	: , : , :
251	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
252	instantiation	265, 266, 318	⊢
	: , : , :
253	instantiation	316, 299, 267	⊢
	: , : , :
254	instantiation	316, 299, 268	⊢
	: , : , :
255	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
256	instantiation	316, 269, 270	⊢
	: , : , :
257	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
258	instantiation	271, 298, 272	⊢
	: , :
259	theorem		⊢
	proveit.numbers.multiplication.mult_complex_closure_bin
260	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.i_is_complex
261	instantiation	273, 274, 275	,  ⊢
	: , :
262	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
263	theorem		⊢
	proveit.physics.quantum.QPE._phase_is_real
264	instantiation	316, 276, 277	⊢
	: , : , :
265	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
266	instantiation	278, 279	⊢
	: , :
267	instantiation	316, 302, 311	⊢
	: , : , :
268	instantiation	316, 280, 281	⊢
	: , : , :
269	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
270	instantiation	316, 282, 283	⊢
	: , : , :
271	theorem		⊢
	proveit.numbers.exponentiation.exp_int_closure
272	instantiation	316, 284, 318	⊢
	: , : , :
273	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
274	instantiation	316, 293, 285	⊢
	: , : , :
275	instantiation	286, 287	,  ⊢
	:
276	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
277	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pi_is_real_pos
278	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
279	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
280	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
281	instantiation	288, 289, 290	⊢
	: , :
282	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
283	instantiation	316, 291, 297	⊢
	: , : , :
284	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
285	instantiation	316, 299, 292	⊢
	: , : , :
286	theorem		⊢
	proveit.numbers.negation.complex_closure
287	instantiation	316, 293, 294	,  ⊢
	: , : , :
288	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
289	instantiation	316, 296, 295	⊢
	: , : , :
290	instantiation	316, 296, 297	⊢
	: , : , :
291	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
292	instantiation	316, 302, 298	⊢
	: , : , :
293	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
294	instantiation	316, 299, 300	,  ⊢
	: , : , :
295	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
296	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
297	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
298	instantiation	316, 314, 301	⊢
	: , : , :
299	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
300	instantiation	316, 302, 303	,  ⊢
	: , : , :
301	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
302	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
303	instantiation	316, 304, 305	,  ⊢
	: , : , :
304	instantiation	306, 307, 308	⊢
	: , :
305	assumption		⊢
306	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
307	instantiation	309, 310, 311	⊢
	: , :
308	theorem		⊢
	proveit.numbers.number_sets.integers.zero_is_int
309	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
310	instantiation	312, 313	⊢
	:
311	instantiation	316, 314, 315	⊢
	: , : , :
312	theorem		⊢
	proveit.numbers.negation.int_closure
313	instantiation	316, 317, 318	⊢
	: , : , :
314	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
315	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
316	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
317	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
318	assumption		⊢
*equality replacement requirements