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