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