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