| step type | requirements | statement |
0 | instantiation | 1, 2, 3, 4, 5, 6 | ⊢ |
| : , : , : , : |
1 | theorem | | ⊢ |
| proveit.linear_algebra.tensors.tensor_prod_is_in_tensor_prod_space |
2 | reference | 183 | ⊢ |
3 | instantiation | 47, 7, 104, 10 | ⊢ |
| : , : , : , : |
4 | instantiation | 8, 172, 173, 72, 18 | ⊢ |
| : , : , : |
5 | instantiation | 47, 9, 104, 10 | ⊢ |
| : , : , : , : |
6 | modus ponens | 11, 12 | ⊢ |
7 | instantiation | 119, 13, 15 | ⊢ |
| : , : , : |
8 | theorem | | ⊢ |
| proveit.logic.booleans.conjunction.redundant_conjunction_general |
9 | instantiation | 119, 14, 15 | ⊢ |
| : , : , : |
10 | instantiation | 68, 16 | ⊢ |
| : , : |
11 | instantiation | 17, 172, 173, 18 | ⊢ |
| : , : , : , : |
12 | generalization | 19 | ⊢ |
13 | instantiation | 20, 21 | ⊢ |
| : , : , : |
14 | instantiation | 20, 21 | ⊢ |
| : , : , : |
15 | instantiation | 110, 22, 23 | ⊢ |
| : , : , : |
16 | instantiation | 24, 25 | ⊢ |
| : , : |
17 | theorem | | ⊢ |
| proveit.logic.booleans.conjunction.conjunction_from_quantification |
18 | instantiation | 88, 26, 27, 115, 28, 29*, 30* | ⊢ |
| : , : , : |
19 | instantiation | 71, 72, 31, 32 | , ⊢ |
| : , : , : , : |
20 | theorem | | ⊢ |
| proveit.core_expr_types.tuples.range_len |
21 | instantiation | 33, 123, 34, 135, 35, 180 | ⊢ |
| : , : |
22 | instantiation | 36, 37 | ⊢ |
| : , : , : |
23 | instantiation | 47, 38, 39, 40 | ⊢ |
| : , : , : , : |
24 | theorem | | ⊢ |
| proveit.core_expr_types.tuples.range_from1_len |
25 | instantiation | 181, 41, 183 | ⊢ |
| : , : , : |
26 | instantiation | 181, 164, 42 | ⊢ |
| : , : , : |
27 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.zero_is_real |
28 | instantiation | 43, 44 | ⊢ |
| : , : |
29 | instantiation | 110, 45, 46 | ⊢ |
| : , : , : |
30 | instantiation | 47, 48, 62, 49 | ⊢ |
| : , : , : , : |
31 | instantiation | 181, 161, 50 | ⊢ |
| : , : , : |
32 | instantiation | 51, 72, 52, 53 | , ⊢ |
| : , : , : , : |
33 | theorem | | ⊢ |
| proveit.numbers.addition.add_nat_closure |
34 | instantiation | 140 | ⊢ |
| : , : , : |
35 | instantiation | 54, 55 | ⊢ |
| : |
36 | axiom | | ⊢ |
| proveit.logic.equality.substitution |
37 | instantiation | 56, 101, 100, 57* | ⊢ |
| : , : |
38 | instantiation | 78, 180, 166, 58, 64, 103, 60, 100 | ⊢ |
| : , : , : , : , : , : |
39 | instantiation | 65, 135, 123, 136, 59, 103, 60, 100 | ⊢ |
| : , : , : , : |
40 | instantiation | 61, 100, 103, 62 | ⊢ |
| : , : , : |
41 | theorem | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat |
42 | instantiation | 181, 167, 172 | ⊢ |
| : , : , : |
43 | theorem | | ⊢ |
| proveit.numbers.ordering.relax_less |
44 | instantiation | 63, 183 | ⊢ |
| : |
45 | instantiation | 78, 180, 166, 135, 79, 136, 64, 101, 100 | ⊢ |
| : , : , : , : , : , : |
46 | instantiation | 65, 135, 166, 136, 79, 101, 100 | ⊢ |
| : , : , : , : |
47 | theorem | | ⊢ |
| proveit.logic.equality.four_chain_transitivity |
48 | instantiation | 110, 66, 67 | ⊢ |
| : , : , : |
49 | instantiation | 68, 69 | ⊢ |
| : , : |
50 | instantiation | 181, 155, 70 | ⊢ |
| : , : , : |
51 | theorem | | ⊢ |
| proveit.linear_algebra.addition.binary_closure |
52 | theorem | | ⊢ |
| proveit.physics.quantum.algebra.ket_zero_in_qubit_space |
53 | instantiation | 71, 72, 73, 74 | , ⊢ |
| : , : , : , : |
54 | theorem | | ⊢ |
| proveit.numbers.negation.nat_closure |
55 | instantiation | 75, 172, 76 | ⊢ |
| : |
56 | theorem | | ⊢ |
| proveit.numbers.negation.distribute_neg_through_binary_sum |
57 | instantiation | 77, 103 | ⊢ |
| : |
58 | instantiation | 148 | ⊢ |
| : , : |
59 | instantiation | 140 | ⊢ |
| : , : , : |
60 | instantiation | 158, 100 | ⊢ |
| : |
61 | theorem | | ⊢ |
| proveit.numbers.addition.subtraction.add_cancel_triple_32 |
62 | instantiation | 116 | ⊢ |
| : |
63 | theorem | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos |
64 | theorem | | ⊢ |
| proveit.numbers.number_sets.complex_numbers.zero_is_complex |
65 | theorem | | ⊢ |
| proveit.numbers.addition.elim_zero_any |
66 | instantiation | 78, 180, 166, 135, 79, 136, 103, 101, 100 | ⊢ |
| : , : , : , : , : , : |
67 | instantiation | 80, 103, 100, 104 | ⊢ |
| : , : , : |
68 | theorem | | ⊢ |
| proveit.logic.equality.equals_reversal |
69 | instantiation | 81, 100 | ⊢ |
| : |
70 | instantiation | 82, 83, 107, 84 | ⊢ |
| : , : |
71 | theorem | | ⊢ |
| proveit.linear_algebra.scalar_multiplication.scalar_mult_closure |
72 | instantiation | 85, 144 | ⊢ |
| : |
73 | instantiation | 152, 86, 87 | , ⊢ |
| : , : |
74 | theorem | | ⊢ |
| proveit.physics.quantum.algebra.ket_one_in_qubit_space |
75 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.nonpos_int_is_int_nonpos |
76 | instantiation | 88, 114, 113, 115, 89, 90*, 91* | ⊢ |
| : , : , : |
77 | theorem | | ⊢ |
| proveit.numbers.negation.double_negation |
78 | theorem | | ⊢ |
| proveit.numbers.addition.disassociation |
79 | instantiation | 148 | ⊢ |
| : , : |
80 | theorem | | ⊢ |
| proveit.numbers.addition.subtraction.add_cancel_triple_12 |
81 | theorem | | ⊢ |
| proveit.numbers.addition.elim_zero_left |
82 | theorem | | ⊢ |
| proveit.numbers.division.div_real_pos_closure |
83 | instantiation | 181, 131, 92 | ⊢ |
| : , : , : |
84 | instantiation | 93, 94 | ⊢ |
| : |
85 | theorem | | ⊢ |
| proveit.linear_algebra.complex_vec_set_is_vec_space |
86 | instantiation | 181, 161, 95 | ⊢ |
| : , : , : |
87 | instantiation | 119, 96, 97 | , ⊢ |
| : , : , : |
88 | theorem | | ⊢ |
| proveit.numbers.addition.weak_bound_via_left_term_bound |
89 | instantiation | 98, 183 | ⊢ |
| : |
90 | instantiation | 99, 100, 101 | ⊢ |
| : , : |
91 | instantiation | 102, 103, 104 | ⊢ |
| : , : |
92 | instantiation | 181, 143, 105 | ⊢ |
| : , : , : |
93 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero |
94 | instantiation | 181, 106, 107 | ⊢ |
| : , : , : |
95 | instantiation | 181, 155, 108 | ⊢ |
| : , : , : |
96 | instantiation | 145, 122, 109 | , ⊢ |
| : , : |
97 | instantiation | 110, 111, 112 | , ⊢ |
| : , : , : |
98 | theorem | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.natural_pos_lower_bound |
99 | theorem | | ⊢ |
| proveit.numbers.addition.commutation |
100 | instantiation | 181, 161, 113 | ⊢ |
| : , : , : |
101 | instantiation | 181, 161, 114 | ⊢ |
| : , : , : |
102 | theorem | | ⊢ |
| proveit.numbers.addition.subtraction.add_cancel_basic |
103 | instantiation | 181, 161, 115 | ⊢ |
| : , : , : |
104 | instantiation | 116 | ⊢ |
| : |
105 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.posnat1 |
106 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero |
107 | instantiation | 117, 118 | ⊢ |
| : |
108 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.e_is_real_pos |
109 | instantiation | 119, 120, 121 | , ⊢ |
| : , : , : |
110 | axiom | | ⊢ |
| proveit.logic.equality.equals_transitivity |
111 | instantiation | 134, 180, 123, 135, 125, 136, 122, 146, 147, 138 | , ⊢ |
| : , : , : , : , : , : |
112 | instantiation | 134, 135, 166, 123, 136, 124, 125, 153, 126, 146, 147, 138 | , ⊢ |
| : , : , : , : , : , : |
113 | instantiation | 181, 164, 127 | ⊢ |
| : , : , : |
114 | instantiation | 181, 164, 128 | ⊢ |
| : , : , : |
115 | instantiation | 129, 130, 183 | ⊢ |
| : , : , : |
116 | axiom | | ⊢ |
| proveit.logic.equality.equals_reflexivity |
117 | theorem | | ⊢ |
| proveit.numbers.exponentiation.sqrt_real_pos_closure |
118 | instantiation | 181, 131, 132 | ⊢ |
| : , : , : |
119 | theorem | | ⊢ |
| proveit.logic.equality.sub_right_side_into |
120 | instantiation | 145, 133, 138 | , ⊢ |
| : , : |
121 | instantiation | 134, 135, 166, 180, 136, 137, 146, 147, 138 | , ⊢ |
| : , : , : , : , : , : |
122 | instantiation | 181, 161, 139 | ⊢ |
| : , : , : |
123 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat3 |
124 | instantiation | 148 | ⊢ |
| : , : |
125 | instantiation | 140 | ⊢ |
| : , : , : |
126 | instantiation | 181, 161, 151 | ⊢ |
| : , : , : |
127 | instantiation | 181, 167, 176 | ⊢ |
| : , : , : |
128 | instantiation | 181, 167, 175 | ⊢ |
| : , : , : |
129 | theorem | | ⊢ |
| proveit.logic.sets.inclusion.unfold_subset_eq |
130 | instantiation | 141, 142 | ⊢ |
| : , : |
131 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos |
132 | instantiation | 181, 143, 144 | ⊢ |
| : , : , : |
133 | instantiation | 145, 146, 147 | , ⊢ |
| : , : |
134 | theorem | | ⊢ |
| proveit.numbers.multiplication.disassociation |
135 | axiom | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.zero_in_nats |
136 | theorem | | ⊢ |
| proveit.core_expr_types.tuples.tuple_len_0_typical_eq |
137 | instantiation | 148 | ⊢ |
| : , : |
138 | instantiation | 181, 161, 149 | ⊢ |
| : , : , : |
139 | instantiation | 150, 157, 151 | ⊢ |
| : , : |
140 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.tuple_len_3_typical_eq |
141 | theorem | | ⊢ |
| proveit.logic.sets.inclusion.relax_proper_subset |
142 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.nat_pos_within_real |
143 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos |
144 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.posnat2 |
145 | theorem | | ⊢ |
| proveit.numbers.multiplication.mult_complex_closure_bin |
146 | theorem | | ⊢ |
| proveit.numbers.number_sets.complex_numbers.i_is_complex |
147 | instantiation | 152, 153, 154 | , ⊢ |
| : , : |
148 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.tuple_len_2_typical_eq |
149 | theorem | | ⊢ |
| proveit.physics.quantum.QPE._phase_is_real |
150 | theorem | | ⊢ |
| proveit.numbers.multiplication.mult_real_closure_bin |
151 | instantiation | 181, 155, 156 | ⊢ |
| : , : , : |
152 | theorem | | ⊢ |
| proveit.numbers.exponentiation.exp_complex_closure |
153 | instantiation | 181, 161, 157 | ⊢ |
| : , : , : |
154 | instantiation | 158, 159 | , ⊢ |
| : |
155 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.real_pos_within_real |
156 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.pi_is_real_pos |
157 | instantiation | 181, 164, 160 | ⊢ |
| : , : , : |
158 | theorem | | ⊢ |
| proveit.numbers.negation.complex_closure |
159 | instantiation | 181, 161, 162 | , ⊢ |
| : , : , : |
160 | instantiation | 181, 167, 163 | ⊢ |
| : , : , : |
161 | theorem | | ⊢ |
| proveit.numbers.number_sets.complex_numbers.real_within_complex |
162 | instantiation | 181, 164, 165 | , ⊢ |
| : , : , : |
163 | instantiation | 181, 179, 166 | ⊢ |
| : , : , : |
164 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.rational_within_real |
165 | instantiation | 181, 167, 168 | , ⊢ |
| : , : , : |
166 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat2 |
167 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.int_within_rational |
168 | instantiation | 181, 169, 170 | , ⊢ |
| : , : , : |
169 | instantiation | 171, 172, 173 | ⊢ |
| : , : |
170 | assumption | | ⊢ |
171 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.int_interval_within_int |
172 | instantiation | 174, 175, 176 | ⊢ |
| : , : |
173 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.zero_is_int |
174 | theorem | | ⊢ |
| proveit.numbers.addition.add_int_closure_bin |
175 | instantiation | 177, 178 | ⊢ |
| : |
176 | instantiation | 181, 179, 180 | ⊢ |
| : , : , : |
177 | theorem | | ⊢ |
| proveit.numbers.negation.int_closure |
178 | instantiation | 181, 182, 183 | ⊢ |
| : , : , : |
179 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.nat_within_int |
180 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat1 |
181 | theorem | | ⊢ |
| proveit.logic.sets.inclusion.superset_membership_from_proper_subset |
182 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.nat_pos_within_int |
183 | assumption | | ⊢ |
*equality replacement requirements |