| step type | requirements | statement |
0 | modus ponens | 1, 2 | ⊢ |
1 | instantiation | 3, 230, 231, 4 | ⊢ |
| : , : , : , : |
2 | generalization | 5 | ⊢ |
3 | theorem | | ⊢ |
| proveit.logic.booleans.conjunction.conjunction_from_quantification |
4 | instantiation | 6, 7, 8, 67, 9, 10*, 11* | ⊢ |
| : , : , : |
5 | instantiation | 168, 12, 13 | , ⊢ |
| : , : , : |
6 | theorem | | ⊢ |
| proveit.numbers.addition.weak_bound_via_left_term_bound |
7 | instantiation | 239, 222, 14 | ⊢ |
| : , : , : |
8 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.zero_is_real |
9 | instantiation | 15, 16 | ⊢ |
| : , : |
10 | instantiation | 168, 17, 18 | ⊢ |
| : , : , : |
11 | instantiation | 62, 19, 20, 21 | ⊢ |
| : , : , : , : |
12 | instantiation | 57, 73, 22, 23, 24*, 25* | , ⊢ |
| : , : , : , : |
13 | instantiation | 168, 26, 27 | ⊢ |
| : , : , : |
14 | instantiation | 239, 225, 230 | ⊢ |
| : , : , : |
15 | theorem | | ⊢ |
| proveit.numbers.ordering.relax_less |
16 | instantiation | 28, 241 | ⊢ |
| : |
17 | instantiation | 44, 238, 224, 191, 45, 192, 97, 46, 118 | ⊢ |
| : , : , : , : , : , : |
18 | instantiation | 29, 191, 224, 192, 45, 46, 118 | ⊢ |
| : , : , : , : |
19 | instantiation | 168, 30, 31 | ⊢ |
| : , : , : |
20 | instantiation | 68 | ⊢ |
| : |
21 | instantiation | 132, 32 | ⊢ |
| : , : |
22 | instantiation | 239, 219, 33 | ⊢ |
| : , : , : |
23 | instantiation | 34, 54, 74, 38 | , ⊢ |
| : , : , : , : |
24 | instantiation | 35, 36 | ⊢ |
| : |
25 | instantiation | 37, 54, 74, 38, 39, 40* | , ⊢ |
| : , : , : , : |
26 | instantiation | 154, 41 | ⊢ |
| : , : , : |
27 | instantiation | 174, 42, 43 | ⊢ |
| : , : , : |
28 | theorem | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos |
29 | theorem | | ⊢ |
| proveit.numbers.addition.elim_zero_any |
30 | instantiation | 44, 238, 224, 191, 45, 192, 48, 46, 118 | ⊢ |
| : , : , : , : , : , : |
31 | instantiation | 47, 48, 118, 49 | ⊢ |
| : , : , : |
32 | instantiation | 50, 118 | ⊢ |
| : |
33 | instantiation | 239, 212, 70 | ⊢ |
| : , : , : |
34 | theorem | | ⊢ |
| proveit.linear_algebra.addition.binary_closure |
35 | theorem | | ⊢ |
| proveit.numbers.absolute_value.abs_non_neg_elim |
36 | instantiation | 51, 52 | ⊢ |
| : |
37 | theorem | | ⊢ |
| proveit.linear_algebra.addition.norm_of_sum_of_orthogonal_pair |
38 | instantiation | 53, 54, 113, 75 | , ⊢ |
| : , : , : , : |
39 | instantiation | 174, 55, 56 | , ⊢ |
| : , : , : |
40 | instantiation | 57, 73, 113, 75, 58* | , ⊢ |
| : , : , : , : |
41 | instantiation | 154, 59 | ⊢ |
| : , : , : |
42 | instantiation | 174, 60, 61 | ⊢ |
| : , : , : |
43 | instantiation | 62, 63, 64, 65 | ⊢ |
| : , : , : , : |
44 | theorem | | ⊢ |
| proveit.numbers.addition.disassociation |
45 | instantiation | 203 | ⊢ |
| : , : |
46 | instantiation | 239, 219, 66 | ⊢ |
| : , : , : |
47 | theorem | | ⊢ |
| proveit.numbers.addition.subtraction.add_cancel_triple_12 |
48 | instantiation | 239, 219, 67 | ⊢ |
| : , : , : |
49 | instantiation | 68 | ⊢ |
| : |
50 | theorem | | ⊢ |
| proveit.numbers.addition.elim_zero_left |
51 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.nonneg_if_in_real_nonneg |
52 | instantiation | 239, 69, 70 | ⊢ |
| : , : , : |
53 | theorem | | ⊢ |
| proveit.linear_algebra.scalar_multiplication.scalar_mult_closure |
54 | instantiation | 71, 173 | ⊢ |
| : |
55 | instantiation | 72, 73, 113, 74, 75 | , ⊢ |
| : , : , : , : , : |
56 | instantiation | 168, 76, 77 | , ⊢ |
| : , : , : |
57 | theorem | | ⊢ |
| proveit.linear_algebra.inner_products.scaled_norm |
58 | instantiation | 168, 78, 79 | , ⊢ |
| : , : , : |
59 | instantiation | 168, 80, 81 | ⊢ |
| : , : , : |
60 | instantiation | 82, 118, 83, 84 | ⊢ |
| : , : , : , : , : |
61 | instantiation | 168, 85, 86 | ⊢ |
| : , : , : |
62 | theorem | | ⊢ |
| proveit.logic.equality.four_chain_transitivity |
63 | instantiation | 154, 87 | ⊢ |
| : , : , : |
64 | instantiation | 154, 87 | ⊢ |
| : , : , : |
65 | instantiation | 159, 118 | ⊢ |
| : |
66 | instantiation | 239, 222, 88 | ⊢ |
| : , : , : |
67 | instantiation | 89, 90, 241 | ⊢ |
| : , : , : |
68 | axiom | | ⊢ |
| proveit.logic.equality.equals_reflexivity |
69 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonneg |
70 | instantiation | 91, 92, 127, 93 | ⊢ |
| : , : |
71 | theorem | | ⊢ |
| proveit.linear_algebra.complex_vec_set_is_vec_space |
72 | theorem | | ⊢ |
| proveit.linear_algebra.inner_products.inner_prod_scalar_mult_right |
73 | instantiation | 94, 173 | ⊢ |
| : |
74 | theorem | | ⊢ |
| proveit.physics.quantum.algebra.ket_zero_in_qubit_space |
75 | theorem | | ⊢ |
| proveit.physics.quantum.algebra.ket_one_in_qubit_space |
76 | instantiation | 154, 95 | ⊢ |
| : , : , : |
77 | instantiation | 96, 113, 97, 98* | , ⊢ |
| : , : |
78 | instantiation | 154, 99 | , ⊢ |
| : , : , : |
79 | instantiation | 140, 100 | ⊢ |
| : |
80 | instantiation | 168, 101, 102 | ⊢ |
| : , : , : |
81 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.add_1_1 |
82 | theorem | | ⊢ |
| proveit.numbers.division.mult_frac_cancel_denom_left |
83 | instantiation | 239, 104, 103 | ⊢ |
| : , : , : |
84 | instantiation | 239, 104, 111 | ⊢ |
| : , : , : |
85 | instantiation | 154, 105 | ⊢ |
| : , : , : |
86 | instantiation | 154, 106 | ⊢ |
| : , : , : |
87 | instantiation | 142, 118 | ⊢ |
| : |
88 | instantiation | 239, 225, 233 | ⊢ |
| : , : , : |
89 | theorem | | ⊢ |
| proveit.logic.sets.inclusion.unfold_subset_eq |
90 | instantiation | 107, 108 | ⊢ |
| : , : |
91 | theorem | | ⊢ |
| proveit.numbers.division.div_real_pos_closure |
92 | instantiation | 239, 161, 109 | ⊢ |
| : , : , : |
93 | instantiation | 110, 111 | ⊢ |
| : |
94 | theorem | | ⊢ |
| proveit.linear_algebra.inner_products.complex_vec_set_is_inner_prod_space |
95 | theorem | | ⊢ |
| proveit.physics.quantum.algebra.ket_zero_and_one_have_zero_inner_prod |
96 | axiom | | ⊢ |
| proveit.linear_algebra.scalar_multiplication.scalar_mult_extends_number_mult |
97 | theorem | | ⊢ |
| proveit.numbers.number_sets.complex_numbers.zero_is_complex |
98 | instantiation | 112, 113 | , ⊢ |
| : |
99 | instantiation | 114, 115, 116* | , ⊢ |
| : |
100 | instantiation | 117, 118, 155 | ⊢ |
| : , : , : |
101 | instantiation | 154, 119 | ⊢ |
| : , : , : |
102 | instantiation | 154, 120 | ⊢ |
| : , : , : |
103 | instantiation | 239, 121, 122 | ⊢ |
| : , : , : |
104 | theorem | | ⊢ |
| proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero |
105 | instantiation | 154, 123 | ⊢ |
| : , : , : |
106 | instantiation | 168, 124, 125 | ⊢ |
| : , : , : |
107 | theorem | | ⊢ |
| proveit.logic.sets.inclusion.relax_proper_subset |
108 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.nat_pos_within_real |
109 | instantiation | 239, 172, 158 | ⊢ |
| : , : , : |
110 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero |
111 | instantiation | 239, 126, 127 | ⊢ |
| : , : , : |
112 | theorem | | ⊢ |
| proveit.numbers.multiplication.mult_zero_right |
113 | instantiation | 209, 128, 129 | , ⊢ |
| : , : |
114 | theorem | | ⊢ |
| proveit.numbers.absolute_value.complex_unit_length |
115 | instantiation | 174, 130, 131 | , ⊢ |
| : , : , : |
116 | instantiation | 132, 133 | , ⊢ |
| : , : |
117 | theorem | | ⊢ |
| proveit.logic.equality.sub_left_side_into |
118 | instantiation | 239, 219, 134 | ⊢ |
| : , : , : |
119 | instantiation | 168, 135, 137 | ⊢ |
| : , : , : |
120 | instantiation | 168, 136, 137 | ⊢ |
| : , : , : |
121 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero |
122 | instantiation | 239, 138, 139 | ⊢ |
| : , : , : |
123 | instantiation | 140, 160 | ⊢ |
| : |
124 | instantiation | 154, 141 | ⊢ |
| : , : , : |
125 | instantiation | 142, 160 | ⊢ |
| : |
126 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero |
127 | instantiation | 143, 144 | ⊢ |
| : |
128 | instantiation | 239, 219, 145 | ⊢ |
| : , : , : |
129 | instantiation | 174, 146, 147 | , ⊢ |
| : , : , : |
130 | instantiation | 205, 195, 148 | , ⊢ |
| : , : |
131 | instantiation | 168, 149, 150 | , ⊢ |
| : , : , : |
132 | theorem | | ⊢ |
| proveit.logic.equality.equals_reversal |
133 | instantiation | 154, 151 | , ⊢ |
| : , : , : |
134 | instantiation | 239, 222, 152 | ⊢ |
| : , : , : |
135 | instantiation | 154, 153 | ⊢ |
| : , : , : |
136 | instantiation | 154, 155 | ⊢ |
| : , : , : |
137 | instantiation | 156, 210 | ⊢ |
| : |
138 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero |
139 | instantiation | 239, 157, 158 | ⊢ |
| : , : , : |
140 | theorem | | ⊢ |
| proveit.numbers.multiplication.elim_one_left |
141 | instantiation | 159, 160 | ⊢ |
| : |
142 | theorem | | ⊢ |
| proveit.numbers.division.frac_one_denom |
143 | theorem | | ⊢ |
| proveit.numbers.exponentiation.sqrt_real_pos_closure |
144 | instantiation | 239, 161, 162 | ⊢ |
| : , : , : |
145 | instantiation | 239, 212, 163 | ⊢ |
| : , : , : |
146 | instantiation | 200, 177, 164 | , ⊢ |
| : , : |
147 | instantiation | 168, 165, 166 | , ⊢ |
| : , : , : |
148 | instantiation | 205, 167, 204 | , ⊢ |
| : , : |
149 | instantiation | 190, 238, 224, 191, 184, 192, 177, 202, 194 | , ⊢ |
| : , : , : , : , : , : |
150 | instantiation | 190, 191, 224, 192, 183, 184, 210, 188, 202, 194 | , ⊢ |
| : , : , : , : , : , : |
151 | instantiation | 168, 169, 170 | , ⊢ |
| : , : , : |
152 | instantiation | 239, 225, 234 | ⊢ |
| : , : , : |
153 | theorem | | ⊢ |
| proveit.physics.quantum.algebra.ket_zero_norm |
154 | axiom | | ⊢ |
| proveit.logic.equality.substitution |
155 | theorem | | ⊢ |
| proveit.physics.quantum.algebra.ket_one_norm |
156 | theorem | | ⊢ |
| proveit.numbers.exponentiation.exponentiated_one |
157 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int |
158 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.posnat1 |
159 | theorem | | ⊢ |
| proveit.numbers.multiplication.elim_one_right |
160 | instantiation | 171, 210 | ⊢ |
| : |
161 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos |
162 | instantiation | 239, 172, 173 | ⊢ |
| : , : , : |
163 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.e_is_real_pos |
164 | instantiation | 174, 175, 176 | , ⊢ |
| : , : , : |
165 | instantiation | 190, 238, 178, 191, 179, 192, 177, 201, 202, 194 | , ⊢ |
| : , : , : , : , : , : |
166 | instantiation | 190, 191, 224, 178, 192, 183, 179, 210, 188, 201, 202, 194 | , ⊢ |
| : , : , : , : , : , : |
167 | instantiation | 180, 215, 181 | , ⊢ |
| : , : |
168 | axiom | | ⊢ |
| proveit.logic.equality.equals_transitivity |
169 | instantiation | 182, 191, 224, 192, 183, 184, 210, 188, 201, 202, 194 | , ⊢ |
| : , : , : , : , : , : , : |
170 | instantiation | 185, 238, 186, 191, 187, 192, 201, 210, 188, 202, 194 | , ⊢ |
| : , : , : , : , : , : |
171 | theorem | | ⊢ |
| proveit.numbers.exponentiation.sqrt_complex_closure |
172 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos |
173 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.posnat2 |
174 | theorem | | ⊢ |
| proveit.logic.equality.sub_right_side_into |
175 | instantiation | 200, 189, 194 | , ⊢ |
| : , : |
176 | instantiation | 190, 191, 224, 238, 192, 193, 201, 202, 194 | , ⊢ |
| : , : , : , : , : , : |
177 | instantiation | 239, 219, 195 | ⊢ |
| : , : , : |
178 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat3 |
179 | instantiation | 196 | ⊢ |
| : , : , : |
180 | theorem | | ⊢ |
| proveit.numbers.exponentiation.exp_real_closure_nat_power |
181 | instantiation | 197, 198 | , ⊢ |
| : |
182 | theorem | | ⊢ |
| proveit.numbers.multiplication.leftward_commutation |
183 | instantiation | 203 | ⊢ |
| : , : |
184 | instantiation | 203 | ⊢ |
| : , : |
185 | theorem | | ⊢ |
| proveit.numbers.multiplication.association |
186 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat4 |
187 | instantiation | 199 | ⊢ |
| : , : , : , : |
188 | instantiation | 239, 219, 206 | ⊢ |
| : , : , : |
189 | instantiation | 200, 201, 202 | , ⊢ |
| : , : |
190 | theorem | | ⊢ |
| proveit.numbers.multiplication.disassociation |
191 | axiom | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.zero_in_nats |
192 | theorem | | ⊢ |
| proveit.core_expr_types.tuples.tuple_len_0_typical_eq |
193 | instantiation | 203 | ⊢ |
| : , : |
194 | instantiation | 239, 219, 204 | ⊢ |
| : , : , : |
195 | instantiation | 205, 215, 206 | ⊢ |
| : , : |
196 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.tuple_len_3_typical_eq |
197 | theorem | | ⊢ |
| proveit.numbers.negation.nat_closure |
198 | instantiation | 207, 226, 208 | , ⊢ |
| : |
199 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.tuple_len_4_typical_eq |
200 | theorem | | ⊢ |
| proveit.numbers.multiplication.mult_complex_closure_bin |
201 | theorem | | ⊢ |
| proveit.numbers.number_sets.complex_numbers.i_is_complex |
202 | instantiation | 209, 210, 211 | , ⊢ |
| : , : |
203 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.tuple_len_2_typical_eq |
204 | theorem | | ⊢ |
| proveit.physics.quantum.QPE._phase_is_real |
205 | theorem | | ⊢ |
| proveit.numbers.multiplication.mult_real_closure_bin |
206 | instantiation | 239, 212, 213 | ⊢ |
| : , : , : |
207 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.nonpos_int_is_int_nonpos |
208 | instantiation | 214, 230, 231, 228 | , ⊢ |
| : , : , : |
209 | theorem | | ⊢ |
| proveit.numbers.exponentiation.exp_complex_closure |
210 | instantiation | 239, 219, 215 | ⊢ |
| : , : , : |
211 | instantiation | 216, 217 | , ⊢ |
| : |
212 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.real_pos_within_real |
213 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.pi_is_real_pos |
214 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.interval_upper_bound |
215 | instantiation | 239, 222, 218 | ⊢ |
| : , : , : |
216 | theorem | | ⊢ |
| proveit.numbers.negation.complex_closure |
217 | instantiation | 239, 219, 220 | , ⊢ |
| : , : , : |
218 | instantiation | 239, 225, 221 | ⊢ |
| : , : , : |
219 | theorem | | ⊢ |
| proveit.numbers.number_sets.complex_numbers.real_within_complex |
220 | instantiation | 239, 222, 223 | , ⊢ |
| : , : , : |
221 | instantiation | 239, 237, 224 | ⊢ |
| : , : , : |
222 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.rational_within_real |
223 | instantiation | 239, 225, 226 | , ⊢ |
| : , : , : |
224 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat2 |
225 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.int_within_rational |
226 | instantiation | 239, 227, 228 | , ⊢ |
| : , : , : |
227 | instantiation | 229, 230, 231 | ⊢ |
| : , : |
228 | assumption | | ⊢ |
229 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.int_interval_within_int |
230 | instantiation | 232, 233, 234 | ⊢ |
| : , : |
231 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.zero_is_int |
232 | theorem | | ⊢ |
| proveit.numbers.addition.add_int_closure_bin |
233 | instantiation | 235, 236 | ⊢ |
| : |
234 | instantiation | 239, 237, 238 | ⊢ |
| : , : , : |
235 | theorem | | ⊢ |
| proveit.numbers.negation.int_closure |
236 | instantiation | 239, 240, 241 | ⊢ |
| : , : , : |
237 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.nat_within_int |
238 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat1 |
239 | theorem | | ⊢ |
| proveit.logic.sets.inclusion.superset_membership_from_proper_subset |
240 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.nat_pos_within_int |
241 | assumption | | ⊢ |
*equality replacement requirements |