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.logic.sets.functions.injections.subset_injection
2	instantiation	4, 120, 5, 215, 6	⊢
	: , : , : , :
3	instantiation	7, 8	⊢
	: , :
4	theorem		⊢
	proveit.numbers.number_sets.integers.interval_subset_eq
5	instantiation	221, 9	⊢
	:
6	instantiation	10, 140, 11, 12, 13, 14	⊢
	: , :
7	theorem		⊢
	proveit.logic.booleans.conjunction.left_from_and
8	instantiation	15, 16	⊢
	: , : , :
9	instantiation	214, 187, 209	⊢
	: , :
10	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_all
11	instantiation	151	⊢
	: , : , :
12	instantiation	17, 18, 19	⊢
	: , :
13	instantiation	20, 21, 166, 22, 23, 24*, 25*	⊢
	: , : , :
14	instantiation	26, 27	⊢
	: , :
15	theorem		⊢
	proveit.logic.sets.functions.bijections.membership_unfolding
16	instantiation	28, 29	⊢
	: , : , :
17	theorem		⊢
	proveit.numbers.ordering.relax_equal_to_less_eq
18	instantiation	223, 210, 30	⊢
	: , : , :
19	instantiation	129	⊢
	:
20	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
21	instantiation	31, 140, 32, 33, 193, 122	⊢
	: , :
22	instantiation	223, 210, 34	⊢
	: , : , :
23	instantiation	35, 209, 208, 199	⊢
	: , : , :
24	instantiation	135, 36, 37, 38	⊢
	: , : , : , :
25	instantiation	147, 39	⊢
	: , :
26	theorem		⊢
	proveit.numbers.ordering.relax_less
27	instantiation	40, 41, 42	⊢
	: , : , :
28	theorem		⊢
	proveit.logic.sets.functions.bijections.elim_domain_condition
29	modus ponens	43, 44	⊢
30	instantiation	223, 218, 120	⊢
	: , : , :
31	theorem		⊢
	proveit.numbers.addition.add_real_closure
32	instantiation	151	⊢
	: , : , :
33	instantiation	223, 210, 45	⊢
	: , : , :
34	instantiation	223, 218, 208	⊢
	: , : , :
35	theorem		⊢
	proveit.numbers.number_sets.integers.interval_upper_bound
36	instantiation	110, 46, 47	⊢
	: , : , :
37	instantiation	129	⊢
	:
38	instantiation	147, 48	⊢
	: , :
39	instantiation	135, 49, 50, 51	⊢
	: , : , : , :
40	axiom		⊢
	proveit.numbers.ordering.transitivity_less_less
41	instantiation	52, 53	⊢
	:
42	instantiation	54, 220	⊢
	:
43	instantiation	55, 56*	⊢
	: , : , : , : , : , :
44	instantiation	57, 58, 59	⊢
	: , :
45	instantiation	223, 218, 133	⊢
	: , : , :
46	instantiation	149, 94	⊢
	: , : , :
47	instantiation	110, 60, 61	⊢
	: , : , :
48	instantiation	149, 109	⊢
	: , : , :
49	instantiation	62, 154, 182	⊢
	: , :
50	instantiation	147, 63	⊢
	: , :
51	instantiation	147, 64	⊢
	: , :
52	theorem		⊢
	proveit.numbers.number_sets.integers.negative_if_in_neg_int
53	instantiation	65, 66	⊢
	:
54	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
55	theorem		⊢
	proveit.logic.sets.functions.bijections.bijection_transitivity
56	instantiation	149, 67	⊢
	: , : , :
57	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
58	instantiation	68, 69, 120, 215, 88	⊢
	: , : , : , : , :
59	theorem		⊢
	proveit.physics.quantum.QPE._sample_space_bijection
60	instantiation	125, 217, 140, 126, 141, 128, 154, 142, 182, 143	⊢
	: , : , : , : , : , :
61	instantiation	113, 126, 225, 128, 70, 154, 142, 182, 107	⊢
	: , : , : , : , : , : , : , :
62	theorem		⊢
	proveit.numbers.negation.distribute_neg_through_binary_sum
63	instantiation	71, 92, 154, 158, 72	⊢
	: , : , :
64	instantiation	110, 73, 74	⊢
	: , : , :
65	theorem		⊢
	proveit.numbers.negation.int_neg_closure
66	instantiation	75, 76, 162	⊢
	: , :
67	instantiation	147, 77	⊢
	: , :
68	theorem		⊢
	proveit.numbers.modular.interval_left_shift_bijection
69	instantiation	78, 225, 206	⊢
	: , :
70	instantiation	144	⊢
	: , :
71	theorem		⊢
	proveit.numbers.addition.subtraction.subtract_from_add
72	instantiation	116, 79, 80	⊢
	: , : , :
73	instantiation	110, 81, 82	⊢
	: , : , :
74	instantiation	110, 83, 84	⊢
	: , : , :
75	theorem		⊢
	proveit.numbers.addition.add_nat_pos_closure_bin
76	instantiation	85, 187, 86	⊢
	:
77	instantiation	87, 88, 89	⊢
	: , :
78	theorem		⊢
	proveit.numbers.exponentiation.exp_natpos_closure
79	instantiation	110, 90, 91	⊢
	: , : , :
80	instantiation	156, 154, 92	⊢
	: , :
81	instantiation	149, 93	⊢
	: , : , :
82	instantiation	149, 94	⊢
	: , : , :
83	instantiation	110, 95, 96	⊢
	: , : , :
84	instantiation	97, 126, 225, 217, 128, 98, 132, 182, 143, 99*	⊢
	: , : , : , : , : , :
85	theorem		⊢
	proveit.numbers.number_sets.integers.pos_int_is_natural_pos
86	instantiation	100, 101, 102	⊢
	: , : , :
87	axiom		⊢
	proveit.physics.quantum.QPE._mod_add_def
88	theorem		⊢
	proveit.physics.quantum.QPE._best_floor_is_int
89	instantiation	223, 103, 104	⊢
	: , : , :
90	instantiation	125, 217, 225, 126, 105, 128, 154, 143, 158	⊢
	: , : , : , : , : , :
91	instantiation	106, 154, 158, 107	⊢
	: , : , :
92	instantiation	223, 203, 108	⊢
	: , : , :
93	instantiation	149, 123	⊢
	: , : , :
94	instantiation	149, 109	⊢
	: , : , :
95	instantiation	110, 111, 112	⊢
	: , : , :
96	instantiation	113, 126, 217, 225, 128, 114, 152, 132, 182, 143, 115	⊢
	: , : , : , : , : , : , : , :
97	theorem		⊢
	proveit.numbers.addition.association
98	instantiation	144	⊢
	: , :
99	instantiation	116, 117, 118	⊢
	: , : , :
100	theorem		⊢
	proveit.numbers.ordering.transitivity_less_less_eq
101	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
102	instantiation	119, 209, 208, 199	⊢
	: , : , :
103	instantiation	207, 120, 215	⊢
	: , :
104	assumption		⊢
105	instantiation	144	⊢
	: , :
106	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_12
107	instantiation	129	⊢
	:
108	instantiation	121, 122, 168	⊢
	: , :
109	instantiation	149, 123	⊢
	: , : , :
110	axiom		⊢
	proveit.logic.equality.equals_transitivity
111	instantiation	125, 126, 225, 217, 128, 127, 152, 132, 124	⊢
	: , : , : , : , : , :
112	instantiation	125, 225, 140, 126, 127, 141, 128, 152, 132, 142, 182, 143	⊢
	: , : , : , : , : , :
113	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_general
114	instantiation	144	⊢
	: , :
115	instantiation	129	⊢
	:
116	theorem		⊢
	proveit.logic.equality.sub_right_side_into
117	instantiation	130, 182, 196, 131	⊢
	: , : , :
118	instantiation	156, 182, 132	⊢
	: , :
119	theorem		⊢
	proveit.numbers.number_sets.integers.interval_lower_bound
120	instantiation	214, 133, 209	⊢
	: , :
121	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
122	instantiation	134, 166	⊢
	:
123	instantiation	135, 136, 137, 138	⊢
	: , : , : , :
124	instantiation	139, 140, 141, 142, 182, 143	⊢
	: , :
125	theorem		⊢
	proveit.numbers.addition.disassociation
126	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
127	instantiation	144	⊢
	: , :
128	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
129	axiom		⊢
	proveit.logic.equality.equals_reflexivity
130	theorem		⊢
	proveit.numbers.addition.subtraction.subtract_from_add_reversed
131	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
132	instantiation	223, 203, 145	⊢
	: , : , :
133	instantiation	221, 215	⊢
	:
134	theorem		⊢
	proveit.numbers.negation.real_closure
135	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
136	instantiation	149, 146	⊢
	: , : , :
137	instantiation	147, 148	⊢
	: , :
138	instantiation	149, 150	⊢
	: , : , :
139	theorem		⊢
	proveit.numbers.addition.add_complex_closure
140	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
141	instantiation	151	⊢
	: , : , :
142	instantiation	153, 152	⊢
	:
143	instantiation	153, 154	⊢
	:
144	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
145	instantiation	223, 210, 155	⊢
	: , : , :
146	instantiation	156, 157, 158	⊢
	: , :
147	theorem		⊢
	proveit.logic.equality.equals_reversal
148	instantiation	159, 196, 168, 167, 184	⊢
	: , : , :
149	axiom		⊢
	proveit.logic.equality.substitution
150	instantiation	160, 161, 162	⊢
	: , :
151	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
152	instantiation	163, 164, 165	⊢
	: , :
153	theorem		⊢
	proveit.numbers.negation.complex_closure
154	instantiation	223, 203, 166	⊢
	: , : , :
155	instantiation	223, 218, 216	⊢
	: , : , :
156	theorem		⊢
	proveit.numbers.addition.commutation
157	instantiation	223, 203, 167	⊢
	: , : , :
158	instantiation	223, 203, 168	⊢
	: , : , :
159	theorem		⊢
	proveit.numbers.exponentiation.product_of_real_powers
160	theorem		⊢
	proveit.numbers.exponentiation.neg_power_as_div
161	instantiation	223, 169, 170	⊢
	: , : , :
162	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
163	theorem		⊢
	proveit.numbers.multiplication.mult_complex_closure_bin
164	instantiation	171, 182, 172, 173	⊢
	: , :
165	instantiation	223, 203, 174	⊢
	: , : , :
166	instantiation	223, 210, 175	⊢
	: , : , :
167	instantiation	176, 177, 213	⊢
	: , : , :
168	instantiation	223, 210, 178	⊢
	: , : , :
169	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
170	instantiation	223, 179, 180	⊢
	: , : , :
171	theorem		⊢
	proveit.numbers.division.div_complex_closure
172	instantiation	181, 196, 182	⊢
	: , :
173	instantiation	183, 184, 185	⊢
	: , : , :
174	instantiation	223, 210, 186	⊢
	: , : , :
175	instantiation	223, 218, 187	⊢
	: , : , :
176	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
177	instantiation	188, 189	⊢
	: , :
178	instantiation	223, 218, 190	⊢
	: , : , :
179	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
180	instantiation	223, 191, 192	⊢
	: , : , :
181	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
182	instantiation	223, 203, 193	⊢
	: , : , :
183	theorem		⊢
	proveit.logic.equality.sub_left_side_into
184	instantiation	194, 201	⊢
	:
185	instantiation	195, 196	⊢
	:
186	instantiation	223, 218, 197	⊢
	: , : , :
187	instantiation	223, 198, 199	⊢
	: , : , :
188	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
189	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
190	instantiation	221, 209	⊢
	:
191	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
192	instantiation	223, 200, 201	⊢
	: , : , :
193	instantiation	223, 210, 202	⊢
	: , : , :
194	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
195	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
196	instantiation	223, 203, 204	⊢
	: , : , :
197	instantiation	205, 222, 206	⊢
	: , :
198	instantiation	207, 209, 208	⊢
	: , :
199	assumption		⊢
200	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
201	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
202	instantiation	223, 218, 209	⊢
	: , : , :
203	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
204	instantiation	223, 210, 211	⊢
	: , : , :
205	theorem		⊢
	proveit.numbers.exponentiation.exp_int_closure
206	instantiation	223, 212, 213	⊢
	: , : , :
207	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
208	instantiation	214, 215, 216	⊢
	: , :
209	instantiation	223, 224, 217	⊢
	: , : , :
210	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
211	instantiation	223, 218, 222	⊢
	: , : , :
212	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
213	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
214	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
215	instantiation	223, 219, 220	⊢
	: , : , :
216	instantiation	221, 222	⊢
	:
217	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
218	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
219	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
220	theorem		⊢
	proveit.physics.quantum.QPE._two_pow_t_minus_one_is_nat_pos
221	theorem		⊢
	proveit.numbers.negation.int_closure
222	instantiation	223, 224, 225	⊢
	: , : , :
223	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
224	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
225	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
*equality replacement requirements