| step type | requirements | statement |
0 | instantiation | 1, 2, 3 | ⊢ |
| : , : , : |
1 | reference | 118 | ⊢ |
2 | instantiation | 118, 4, 5 | ⊢ |
| : , : , : |
3 | instantiation | 6, 26, 55, 7, 24, 8* | ⊢ |
| : , : , : |
4 | instantiation | 118, 9, 10 | ⊢ |
| : , : , : |
5 | instantiation | 11, 187, 182, 55, 23, 24 | ⊢ |
| : , : , : , : , : |
6 | theorem | | ⊢ |
| proveit.physics.quantum.algebra.scalar_mult_factorization |
7 | instantiation | 114 | ⊢ |
| : , : , : |
8 | instantiation | 12, 55, 13 | ⊢ |
| : , : |
9 | instantiation | 14, 55, 187, 109, 110, 15 | ⊢ |
| : , : , : , : , : , : |
10 | instantiation | 22, 182, 176, 109, 42, 23, 110, 16 | ⊢ |
| : , : , : , : , : , : |
11 | theorem | | ⊢ |
| proveit.physics.quantum.algebra.qmult_pulling_scalar_out_front |
12 | axiom | | ⊢ |
| proveit.linear_algebra.scalar_multiplication.scalar_mult_extends_number_mult |
13 | instantiation | 78, 17, 29 | ⊢ |
| : , : , : |
14 | theorem | | ⊢ |
| proveit.physics.quantum.algebra.scalar_mult_absorption |
15 | instantiation | 78, 24, 18 | ⊢ |
| : , : , : |
16 | instantiation | 78, 19, 20 | ⊢ |
| : , : , : |
17 | instantiation | 21, 67, 68, 32, 33 | ⊢ |
| : , : , : , : |
18 | instantiation | 22, 187, 176, 109, 23, 110, 24 | ⊢ |
| : , : , : , : , : , : |
19 | instantiation | 31, 67, 68, 25, 33 | ⊢ |
| : , : , : , : |
20 | instantiation | 41, 26, 27 | ⊢ |
| : , : , : |
21 | theorem | | ⊢ |
| proveit.physics.quantum.algebra.qmult_op_ket_is_ket |
22 | theorem | | ⊢ |
| proveit.physics.quantum.algebra.qmult_disassociation |
23 | instantiation | 124 | ⊢ |
| : , : |
24 | instantiation | 78, 28, 29 | ⊢ |
| : , : , : |
25 | instantiation | 66, 67, 68, 30 | ⊢ |
| : , : , : |
26 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.posnat3 |
27 | instantiation | 114 | ⊢ |
| : , : , : |
28 | instantiation | 31, 67, 68, 32, 33 | ⊢ |
| : , : , : , : |
29 | instantiation | 41, 176, 34 | ⊢ |
| : , : , : |
30 | instantiation | 78, 35, 36 | ⊢ |
| : , : , : |
31 | theorem | | ⊢ |
| proveit.physics.quantum.algebra.qmult_op_ket_in_QmultCodomain |
32 | instantiation | 66, 67, 68, 37 | ⊢ |
| : , : , : |
33 | modus ponens | 38, 39 | ⊢ |
34 | instantiation | 124 | ⊢ |
| : , : |
35 | instantiation | 43, 67, 68, 40, 44 | ⊢ |
| : , : , : , : , : |
36 | instantiation | 41, 176, 42 | ⊢ |
| : , : , : |
37 | instantiation | 43, 67, 68, 56, 44 | ⊢ |
| : , : , : , : , : |
38 | instantiation | 45, 174, 51 | ⊢ |
| : , : , : , : , : , : |
39 | generalization | 46 | ⊢ |
40 | instantiation | 66, 67, 68, 47 | ⊢ |
| : , : , : |
41 | axiom | | ⊢ |
| proveit.physics.quantum.algebra.multi_qmult_def |
42 | instantiation | 124 | ⊢ |
| : , : |
43 | theorem | | ⊢ |
| proveit.physics.quantum.algebra.qmult_op_op_is_op |
44 | instantiation | 48, 82, 49 | ⊢ |
| : , : , : |
45 | theorem | | ⊢ |
| proveit.linear_algebra.addition.summation_closure |
46 | instantiation | 50, 51, 52, 53 | ⊢ |
| : , : , : , : |
47 | instantiation | 54, 55, 67, 68, 56 | ⊢ |
| : , : , : , : |
48 | theorem | | ⊢ |
| proveit.physics.quantum.algebra.qmult_matrix_is_linmap |
49 | instantiation | 146, 57, 58 | ⊢ |
| : , : , : |
50 | theorem | | ⊢ |
| proveit.linear_algebra.scalar_multiplication.scalar_mult_closure |
51 | instantiation | 59, 82 | ⊢ |
| : |
52 | instantiation | 76, 60, 61 | ⊢ |
| : , : |
53 | instantiation | 62, 184, 150 | ⊢ |
| : , : |
54 | theorem | | ⊢ |
| proveit.physics.quantum.algebra.qmult_op_complex_closure |
55 | instantiation | 90, 63, 64, 65 | ⊢ |
| : , : |
56 | instantiation | 66, 67, 68, 69 | ⊢ |
| : , : , : |
57 | instantiation | 70, 82 | ⊢ |
| : |
58 | instantiation | 71, 184 | ⊢ |
| : |
59 | theorem | | ⊢ |
| proveit.linear_algebra.complex_vec_set_is_vec_space |
60 | instantiation | 185, 153, 72 | ⊢ |
| : , : , : |
61 | instantiation | 95, 73, 74 | ⊢ |
| : , : , : |
62 | theorem | | ⊢ |
| proveit.physics.quantum.algebra.num_ket_in_register_space |
63 | instantiation | 185, 153, 75 | ⊢ |
| : , : , : |
64 | instantiation | 76, 145, 77 | ⊢ |
| : , : |
65 | instantiation | 78, 79, 80 | ⊢ |
| : , : , : |
66 | theorem | | ⊢ |
| proveit.physics.quantum.algebra.qmult_op_is_linmap |
67 | instantiation | 81, 82 | ⊢ |
| : |
68 | theorem | | ⊢ |
| proveit.linear_algebra.inner_products.complex_set_is_hilbert_space |
69 | instantiation | 83, 184, 84 | ⊢ |
| : , : |
70 | theorem | | ⊢ |
| proveit.linear_algebra.matrices.unitaries_are_matrices |
71 | theorem | | ⊢ |
| proveit.physics.quantum.QFT.invFT_is_unitary |
72 | instantiation | 185, 134, 85 | ⊢ |
| : , : , : |
73 | instantiation | 121, 98, 86 | ⊢ |
| : , : |
74 | instantiation | 118, 87, 88 | ⊢ |
| : , : , : |
75 | instantiation | 185, 164, 89 | ⊢ |
| : , : , : |
76 | theorem | | ⊢ |
| proveit.numbers.exponentiation.exp_complex_closure |
77 | instantiation | 90, 130, 145, 105 | ⊢ |
| : , : |
78 | theorem | | ⊢ |
| proveit.logic.equality.sub_left_side_into |
79 | instantiation | 91, 152, 92 | ⊢ |
| : , : |
80 | instantiation | 127, 93 | ⊢ |
| : , : , : |
81 | theorem | | ⊢ |
| proveit.linear_algebra.inner_products.complex_vec_set_is_hilbert_space |
82 | instantiation | 94, 182, 179 | ⊢ |
| : , : |
83 | theorem | | ⊢ |
| proveit.physics.quantum.algebra.num_bra_is_lin_map |
84 | assumption | | ⊢ |
85 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.e_is_real_pos |
86 | instantiation | 95, 96, 97 | ⊢ |
| : , : , : |
87 | instantiation | 108, 187, 99, 109, 101, 110, 98, 122, 123, 112 | ⊢ |
| : , : , : , : , : , : |
88 | instantiation | 108, 109, 182, 99, 110, 100, 101, 145, 113, 122, 123, 112 | ⊢ |
| : , : , : , : , : , : |
89 | instantiation | 185, 173, 181 | ⊢ |
| : , : , : |
90 | theorem | | ⊢ |
| proveit.numbers.division.div_complex_closure |
91 | theorem | | ⊢ |
| proveit.numbers.exponentiation.exp_rational_non_zero__not_zero |
92 | instantiation | 185, 102, 103 | ⊢ |
| : , : , : |
93 | instantiation | 104, 130, 145, 105, 106* | ⊢ |
| : , : |
94 | theorem | | ⊢ |
| proveit.numbers.exponentiation.exp_natpos_closure |
95 | theorem | | ⊢ |
| proveit.logic.equality.sub_right_side_into |
96 | instantiation | 121, 107, 112 | ⊢ |
| : , : |
97 | instantiation | 108, 109, 182, 187, 110, 111, 122, 123, 112 | ⊢ |
| : , : , : , : , : , : |
98 | instantiation | 121, 145, 113 | ⊢ |
| : , : |
99 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat3 |
100 | instantiation | 124 | ⊢ |
| : , : |
101 | instantiation | 114 | ⊢ |
| : , : , : |
102 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational_nonzero |
103 | instantiation | 115, 158, 116 | ⊢ |
| : , : |
104 | theorem | | ⊢ |
| proveit.numbers.division.div_as_mult |
105 | instantiation | 117, 176 | ⊢ |
| : |
106 | instantiation | 118, 119, 120 | ⊢ |
| : , : , : |
107 | instantiation | 121, 122, 123 | ⊢ |
| : , : |
108 | theorem | | ⊢ |
| proveit.numbers.multiplication.disassociation |
109 | axiom | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.zero_in_nats |
110 | theorem | | ⊢ |
| proveit.core_expr_types.tuples.tuple_len_0_typical_eq |
111 | instantiation | 124 | ⊢ |
| : , : |
112 | instantiation | 185, 153, 125 | ⊢ |
| : , : , : |
113 | instantiation | 185, 153, 126 | ⊢ |
| : , : , : |
114 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.tuple_len_3_typical_eq |
115 | theorem | | ⊢ |
| proveit.numbers.multiplication.mult_rational_pos_closure_bin |
116 | instantiation | 185, 175, 184 | ⊢ |
| : , : , : |
117 | theorem | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos |
118 | axiom | | ⊢ |
| proveit.logic.equality.equals_transitivity |
119 | instantiation | 127, 128 | ⊢ |
| : , : , : |
120 | instantiation | 129, 130, 131 | ⊢ |
| : , : |
121 | theorem | | ⊢ |
| proveit.numbers.multiplication.mult_complex_closure_bin |
122 | theorem | | ⊢ |
| proveit.numbers.number_sets.complex_numbers.i_is_complex |
123 | instantiation | 185, 153, 132 | ⊢ |
| : , : , : |
124 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.tuple_len_2_typical_eq |
125 | instantiation | 185, 164, 133 | ⊢ |
| : , : , : |
126 | instantiation | 185, 134, 135 | ⊢ |
| : , : , : |
127 | axiom | | ⊢ |
| proveit.logic.equality.substitution |
128 | instantiation | 136, 137, 174, 138* | ⊢ |
| : , : |
129 | theorem | | ⊢ |
| proveit.numbers.multiplication.commutation |
130 | instantiation | 185, 153, 139 | ⊢ |
| : , : , : |
131 | instantiation | 185, 153, 140 | ⊢ |
| : , : , : |
132 | theorem | | ⊢ |
| proveit.physics.quantum.QPE._phase_is_real |
133 | instantiation | 185, 173, 141 | ⊢ |
| : , : , : |
134 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.real_pos_within_real |
135 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.pi_is_real_pos |
136 | theorem | | ⊢ |
| proveit.numbers.exponentiation.neg_power_as_div |
137 | instantiation | 185, 142, 143 | ⊢ |
| : , : , : |
138 | instantiation | 144, 145 | ⊢ |
| : |
139 | instantiation | 146, 147, 184 | ⊢ |
| : , : , : |
140 | instantiation | 185, 164, 148 | ⊢ |
| : , : , : |
141 | instantiation | 185, 149, 150 | ⊢ |
| : , : , : |
142 | theorem | | ⊢ |
| proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero |
143 | instantiation | 185, 151, 152 | ⊢ |
| : , : , : |
144 | theorem | | ⊢ |
| proveit.numbers.exponentiation.complex_x_to_first_power_is_x |
145 | instantiation | 185, 153, 154 | ⊢ |
| : , : , : |
146 | theorem | | ⊢ |
| proveit.logic.sets.inclusion.unfold_subset_eq |
147 | instantiation | 155, 156 | ⊢ |
| : , : |
148 | instantiation | 185, 157, 158 | ⊢ |
| : , : , : |
149 | instantiation | 159, 160, 161 | ⊢ |
| : , : |
150 | assumption | | ⊢ |
151 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero |
152 | instantiation | 185, 162, 163 | ⊢ |
| : , : , : |
153 | theorem | | ⊢ |
| proveit.numbers.number_sets.complex_numbers.real_within_complex |
154 | instantiation | 185, 164, 165 | ⊢ |
| : , : , : |
155 | theorem | | ⊢ |
| proveit.logic.sets.inclusion.relax_proper_subset |
156 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.nat_pos_within_real |
157 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational |
158 | instantiation | 166, 167, 168 | ⊢ |
| : , : |
159 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.int_interval_within_int |
160 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.zero_is_int |
161 | instantiation | 169, 170, 171 | ⊢ |
| : , : |
162 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero |
163 | instantiation | 185, 172, 176 | ⊢ |
| : , : , : |
164 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.rational_within_real |
165 | instantiation | 185, 173, 178 | ⊢ |
| : , : , : |
166 | theorem | | ⊢ |
| proveit.numbers.division.div_rational_pos_closure |
167 | instantiation | 185, 175, 174 | ⊢ |
| : , : , : |
168 | instantiation | 185, 175, 176 | ⊢ |
| : , : , : |
169 | theorem | | ⊢ |
| proveit.numbers.addition.add_int_closure_bin |
170 | instantiation | 177, 178, 179 | ⊢ |
| : , : |
171 | instantiation | 180, 181 | ⊢ |
| : |
172 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int |
173 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.int_within_rational |
174 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.posnat1 |
175 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos |
176 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.posnat2 |
177 | theorem | | ⊢ |
| proveit.numbers.exponentiation.exp_int_closure |
178 | instantiation | 185, 186, 182 | ⊢ |
| : , : , : |
179 | instantiation | 185, 183, 184 | ⊢ |
| : , : , : |
180 | theorem | | ⊢ |
| proveit.numbers.negation.int_closure |
181 | instantiation | 185, 186, 187 | ⊢ |
| : , : , : |
182 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat2 |
183 | theorem | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat |
184 | axiom | | ⊢ |
| proveit.physics.quantum.QPE._t_in_natural_pos |
185 | theorem | | ⊢ |
| proveit.logic.sets.inclusion.superset_membership_from_proper_subset |
186 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.nat_within_int |
187 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat1 |
*equality replacement requirements |