| step type | requirements | statement |
0 | instantiation | 1, 2, 3, 4, 5, 6 | ⊢ |
| : , : |
1 | theorem | | ⊢ |
| proveit.logic.booleans.conjunction.and_if_all |
2 | reference | 181 | ⊢ |
3 | instantiation | 192 | ⊢ |
| : , : , : |
4 | instantiation | 7, 8, 9 | ⊢ |
| : , : , : |
5 | instantiation | 60, 227, 206, 10, 11, 12* | ⊢ |
| : , : , : |
6 | instantiation | 13, 14, 15 | ⊢ |
| : , : |
7 | theorem | | ⊢ |
| proveit.numbers.ordering.transitivity_less_eq_less_eq |
8 | instantiation | 60, 16, 206, 25, 26, 17*, 18* | ⊢ |
| : , : , : |
9 | instantiation | 60, 90, 19, 20, 21, 22*, 23* | ⊢ |
| : , : , : |
10 | instantiation | 110, 25, 227 | ⊢ |
| : , : |
11 | instantiation | 24, 206, 25, 227, 26, 27 | ⊢ |
| : , : , : |
12 | instantiation | 176, 28, 159, 29 | ⊢ |
| : , : , : , : |
13 | theorem | | ⊢ |
| proveit.numbers.ordering.relax_equal_to_less_eq |
14 | instantiation | 231, 232, 254 | ⊢ |
| : , : , : |
15 | instantiation | 172 | ⊢ |
| : |
16 | instantiation | 30, 181, 31, 32, 227, 111 | ⊢ |
| : , : |
17 | instantiation | 176, 33, 34, 35 | ⊢ |
| : , : , : , : |
18 | instantiation | 188, 36 | ⊢ |
| : , : |
19 | instantiation | 128, 240, 61 | ⊢ |
| : , : |
20 | instantiation | 110, 112, 240 | ⊢ |
| : , : |
21 | instantiation | 37, 38, 39, 256, 40, 41 | ⊢ |
| : , : , : |
22 | instantiation | 188, 42 | ⊢ |
| : , : |
23 | instantiation | 176, 43, 44, 45 | ⊢ |
| : , : , : , : |
24 | theorem | | ⊢ |
| proveit.numbers.ordering.less_eq_add_right |
25 | instantiation | 257, 246, 46 | ⊢ |
| : , : , : |
26 | instantiation | 47, 245, 244, 235 | ⊢ |
| : , : , : |
27 | instantiation | 83, 251 | ⊢ |
| : |
28 | instantiation | 154, 48, 49 | ⊢ |
| : , : , : |
29 | instantiation | 188, 166 | ⊢ |
| : , : |
30 | theorem | | ⊢ |
| proveit.numbers.addition.add_real_closure |
31 | instantiation | 192 | ⊢ |
| : , : , : |
32 | instantiation | 257, 246, 50 | ⊢ |
| : , : , : |
33 | instantiation | 154, 51, 52 | ⊢ |
| : , : , : |
34 | instantiation | 172 | ⊢ |
| : |
35 | instantiation | 188, 53 | ⊢ |
| : , : |
36 | instantiation | 176, 65, 54, 55 | ⊢ |
| : , : , : , : |
37 | theorem | | ⊢ |
| proveit.numbers.ordering.less_add_right_weak_int |
38 | instantiation | 257, 258, 56 | ⊢ |
| : , : , : |
39 | instantiation | 257, 258, 57 | ⊢ |
| : , : , : |
40 | instantiation | 58, 240, 61, 206, 123, 59 | ⊢ |
| : , : , : |
41 | instantiation | 60, 199, 61, 107, 62, 63*, 64* | ⊢ |
| : , : , : |
42 | instantiation | 176, 65, 66, 67 | ⊢ |
| : , : , : , : |
43 | instantiation | 168, 169, 259, 251, 171, 69, 93, 230, 68 | ⊢ |
| : , : , : , : , : , : |
44 | instantiation | 168, 259, 169, 69, 147, 171, 93, 230, 184, 197 | ⊢ |
| : , : , : , : , : , : |
45 | instantiation | 176, 70, 71, 72 | ⊢ |
| : , : , : , : |
46 | instantiation | 257, 252, 244 | ⊢ |
| : , : , : |
47 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.interval_upper_bound |
48 | instantiation | 190, 73 | ⊢ |
| : , : , : |
49 | instantiation | 154, 74, 75 | ⊢ |
| : , : , : |
50 | instantiation | 257, 252, 76 | ⊢ |
| : , : , : |
51 | instantiation | 190, 134 | ⊢ |
| : , : , : |
52 | instantiation | 154, 77, 78 | ⊢ |
| : , : , : |
53 | instantiation | 190, 153 | ⊢ |
| : , : , : |
54 | instantiation | 196, 184, 197 | ⊢ |
| : , : |
55 | instantiation | 188, 79 | ⊢ |
| : , : |
56 | instantiation | 80, 259, 169 | ⊢ |
| : , : |
57 | instantiation | 80, 259, 81 | ⊢ |
| : , : |
58 | theorem | | ⊢ |
| proveit.numbers.multiplication.strong_bound_via_right_factor_bound |
59 | instantiation | 82, 237 | ⊢ |
| : |
60 | theorem | | ⊢ |
| proveit.numbers.addition.weak_bound_via_left_term_bound |
61 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.zero_is_real |
62 | instantiation | 83, 181 | ⊢ |
| : |
63 | instantiation | 160, 84, 85 | ⊢ |
| : , : , : |
64 | instantiation | 165, 230, 219, 86, 87 | ⊢ |
| : , : , : |
65 | instantiation | 88, 194, 219 | ⊢ |
| : , : |
66 | instantiation | 172 | ⊢ |
| : |
67 | instantiation | 188, 89 | ⊢ |
| : , : |
68 | instantiation | 257, 239, 90 | ⊢ |
| : , : , : |
69 | instantiation | 185 | ⊢ |
| : , : |
70 | instantiation | 91, 251, 93, 230, 184, 197 | ⊢ |
| : , : , : , : , : , : , : |
71 | instantiation | 137, 169, 259, 171, 92, 150, 93, 184, 230, 197, 94* | ⊢ |
| : , : , : , : , : , : |
72 | instantiation | 137, 251, 259, 169, 150, 171, 194, 230, 197, 151* | ⊢ |
| : , : , : , : , : , : |
73 | instantiation | 154, 95, 96 | ⊢ |
| : , : , : |
74 | instantiation | 168, 169, 259, 251, 171, 97, 193, 197, 219 | ⊢ |
| : , : , : , : , : , : |
75 | instantiation | 98, 219, 193, 99 | ⊢ |
| : , : , : |
76 | instantiation | 255, 249 | ⊢ |
| : |
77 | instantiation | 168, 251, 181, 169, 182, 171, 194, 183, 219, 184 | ⊢ |
| : , : , : , : , : , : |
78 | instantiation | 157, 169, 259, 171, 100, 194, 183, 219, 101 | ⊢ |
| : , : , : , : , : , : , : , : |
79 | instantiation | 154, 102, 103 | ⊢ |
| : , : , : |
80 | theorem | | ⊢ |
| proveit.numbers.multiplication.mult_nat_closure_bin |
81 | instantiation | 104, 224, 105 | ⊢ |
| : |
82 | theorem | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos |
83 | theorem | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.natural_lower_bound |
84 | instantiation | 106, 197 | ⊢ |
| : |
85 | instantiation | 196, 197, 145 | ⊢ |
| : , : |
86 | instantiation | 257, 239, 107 | ⊢ |
| : , : , : |
87 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.add_2_1 |
88 | theorem | | ⊢ |
| proveit.numbers.negation.distribute_neg_through_binary_sum |
89 | instantiation | 154, 108, 109 | ⊢ |
| : , : , : |
90 | instantiation | 110, 111, 199 | ⊢ |
| : , : |
91 | theorem | | ⊢ |
| proveit.numbers.addition.leftward_commutation |
92 | instantiation | 185 | ⊢ |
| : , : |
93 | instantiation | 257, 239, 112 | ⊢ |
| : , : , : |
94 | instantiation | 154, 113, 114, 115* | ⊢ |
| : , : , : |
95 | instantiation | 190, 133 | ⊢ |
| : , : , : |
96 | instantiation | 154, 116, 117 | ⊢ |
| : , : , : |
97 | instantiation | 185 | ⊢ |
| : , : |
98 | theorem | | ⊢ |
| proveit.numbers.addition.subtraction.add_cancel_triple_32 |
99 | instantiation | 172 | ⊢ |
| : |
100 | instantiation | 185 | ⊢ |
| : , : |
101 | instantiation | 172 | ⊢ |
| : |
102 | instantiation | 154, 118, 119 | ⊢ |
| : , : , : |
103 | instantiation | 154, 120, 121 | ⊢ |
| : , : , : |
104 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.nonneg_int_is_natural |
105 | instantiation | 122, 123 | ⊢ |
| : , : |
106 | theorem | | ⊢ |
| proveit.numbers.addition.elim_zero_right |
107 | instantiation | 257, 246, 124 | ⊢ |
| : , : , : |
108 | instantiation | 190, 125 | ⊢ |
| : , : , : |
109 | instantiation | 154, 126, 127 | ⊢ |
| : , : , : |
110 | theorem | | ⊢ |
| proveit.numbers.addition.add_real_closure_bin |
111 | instantiation | 208, 206 | ⊢ |
| : |
112 | instantiation | 128, 240, 206 | ⊢ |
| : , : |
113 | instantiation | 190, 129 | ⊢ |
| : , : , : |
114 | instantiation | 188, 130 | ⊢ |
| : , : |
115 | instantiation | 154, 131, 132 | ⊢ |
| : , : , : |
116 | instantiation | 168, 169, 259, 251, 171, 170, 193, 175, 219 | ⊢ |
| : , : , : , : , : , : |
117 | instantiation | 137, 251, 259, 169, 138, 171, 193, 175, 219, 139* | ⊢ |
| : , : , : , : , : , : |
118 | instantiation | 190, 133 | ⊢ |
| : , : , : |
119 | instantiation | 190, 134 | ⊢ |
| : , : , : |
120 | instantiation | 154, 135, 136 | ⊢ |
| : , : , : |
121 | instantiation | 137, 169, 259, 251, 171, 138, 175, 219, 184, 139* | ⊢ |
| : , : , : , : , : , : |
122 | theorem | | ⊢ |
| proveit.numbers.ordering.relax_less |
123 | instantiation | 140, 141, 142 | ⊢ |
| : , : , : |
124 | instantiation | 257, 252, 143 | ⊢ |
| : , : , : |
125 | instantiation | 144, 230 | ⊢ |
| : |
126 | instantiation | 168, 251, 259, 169, 147, 171, 145, 184, 197 | ⊢ |
| : , : , : , : , : , : |
127 | instantiation | 146, 169, 259, 171, 147, 184, 197 | ⊢ |
| : , : , : , : |
128 | theorem | | ⊢ |
| proveit.numbers.multiplication.mult_real_closure_bin |
129 | instantiation | 188, 148 | ⊢ |
| : , : |
130 | instantiation | 149, 169, 259, 251, 171, 150, 230, 197, 194 | ⊢ |
| : , : , : , : , : , : |
131 | instantiation | 190, 151 | ⊢ |
| : , : , : |
132 | instantiation | 152, 194 | ⊢ |
| : |
133 | instantiation | 190, 166 | ⊢ |
| : , : , : |
134 | instantiation | 190, 153 | ⊢ |
| : , : , : |
135 | instantiation | 154, 155, 156 | ⊢ |
| : , : , : |
136 | instantiation | 157, 169, 251, 259, 171, 158, 193, 175, 219, 184, 159 | ⊢ |
| : , : , : , : , : , : , : , : |
137 | theorem | | ⊢ |
| proveit.numbers.addition.association |
138 | instantiation | 185 | ⊢ |
| : , : |
139 | instantiation | 160, 161, 162 | ⊢ |
| : , : , : |
140 | theorem | | ⊢ |
| proveit.numbers.ordering.transitivity_less_less_eq |
141 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.less_0_1 |
142 | instantiation | 163, 245, 244, 235 | ⊢ |
| : , : , : |
143 | instantiation | 257, 258, 181 | ⊢ |
| : , : , : |
144 | theorem | | ⊢ |
| proveit.numbers.multiplication.mult_zero_right |
145 | theorem | | ⊢ |
| proveit.numbers.number_sets.complex_numbers.zero_is_complex |
146 | theorem | | ⊢ |
| proveit.numbers.addition.elim_zero_any |
147 | instantiation | 185 | ⊢ |
| : , : |
148 | instantiation | 164, 194 | ⊢ |
| : |
149 | theorem | | ⊢ |
| proveit.numbers.multiplication.distribute_through_sum |
150 | instantiation | 185 | ⊢ |
| : , : |
151 | instantiation | 165, 219, 230, 174 | ⊢ |
| : , : , : |
152 | theorem | | ⊢ |
| proveit.numbers.multiplication.elim_one_left |
153 | instantiation | 190, 166 | ⊢ |
| : , : , : |
154 | axiom | | ⊢ |
| proveit.logic.equality.equals_transitivity |
155 | instantiation | 168, 169, 259, 251, 171, 170, 193, 175, 167 | ⊢ |
| : , : , : , : , : , : |
156 | instantiation | 168, 259, 181, 169, 170, 182, 171, 193, 175, 183, 219, 184 | ⊢ |
| : , : , : , : , : , : |
157 | theorem | | ⊢ |
| proveit.numbers.addition.subtraction.add_cancel_general |
158 | instantiation | 185 | ⊢ |
| : , : |
159 | instantiation | 172 | ⊢ |
| : |
160 | theorem | | ⊢ |
| proveit.logic.equality.sub_right_side_into |
161 | instantiation | 173, 219, 230, 174 | ⊢ |
| : , : , : |
162 | instantiation | 196, 219, 175 | ⊢ |
| : , : |
163 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.interval_lower_bound |
164 | theorem | | ⊢ |
| proveit.numbers.negation.mult_neg_one_left |
165 | theorem | | ⊢ |
| proveit.numbers.addition.subtraction.subtract_from_add |
166 | instantiation | 176, 177, 178, 179 | ⊢ |
| : , : , : , : |
167 | instantiation | 180, 181, 182, 183, 219, 184 | ⊢ |
| : , : |
168 | theorem | | ⊢ |
| proveit.numbers.addition.disassociation |
169 | axiom | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.zero_in_nats |
170 | instantiation | 185 | ⊢ |
| : , : |
171 | theorem | | ⊢ |
| proveit.core_expr_types.tuples.tuple_len_0_typical_eq |
172 | axiom | | ⊢ |
| proveit.logic.equality.equals_reflexivity |
173 | theorem | | ⊢ |
| proveit.numbers.addition.subtraction.subtract_from_add_reversed |
174 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.add_1_1 |
175 | instantiation | 257, 239, 186 | ⊢ |
| : , : , : |
176 | theorem | | ⊢ |
| proveit.logic.equality.four_chain_transitivity |
177 | instantiation | 190, 187 | ⊢ |
| : , : , : |
178 | instantiation | 188, 189 | ⊢ |
| : , : |
179 | instantiation | 190, 191 | ⊢ |
| : , : , : |
180 | theorem | | ⊢ |
| proveit.numbers.addition.add_complex_closure |
181 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat3 |
182 | instantiation | 192 | ⊢ |
| : , : , : |
183 | instantiation | 207, 193 | ⊢ |
| : |
184 | instantiation | 207, 194 | ⊢ |
| : |
185 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.tuple_len_2_typical_eq |
186 | instantiation | 257, 246, 195 | ⊢ |
| : , : , : |
187 | instantiation | 196, 214, 197 | ⊢ |
| : , : |
188 | theorem | | ⊢ |
| proveit.logic.equality.equals_reversal |
189 | instantiation | 198, 230, 199, 223, 221 | ⊢ |
| : , : , : |
190 | axiom | | ⊢ |
| proveit.logic.equality.substitution |
191 | instantiation | 200, 201, 202 | ⊢ |
| : , : |
192 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.tuple_len_3_typical_eq |
193 | instantiation | 203, 204, 205 | ⊢ |
| : , : |
194 | instantiation | 257, 239, 206 | ⊢ |
| : , : , : |
195 | instantiation | 257, 252, 250 | ⊢ |
| : , : , : |
196 | theorem | | ⊢ |
| proveit.numbers.addition.commutation |
197 | instantiation | 207, 219 | ⊢ |
| : |
198 | theorem | | ⊢ |
| proveit.numbers.exponentiation.product_of_real_powers |
199 | instantiation | 208, 227 | ⊢ |
| : |
200 | theorem | | ⊢ |
| proveit.numbers.exponentiation.neg_power_as_div |
201 | instantiation | 257, 209, 210 | ⊢ |
| : , : , : |
202 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.posnat1 |
203 | theorem | | ⊢ |
| proveit.numbers.multiplication.mult_complex_closure_bin |
204 | instantiation | 211, 219, 212, 213 | ⊢ |
| : , : |
205 | instantiation | 218, 230, 214 | ⊢ |
| : , : |
206 | instantiation | 257, 246, 215 | ⊢ |
| : , : , : |
207 | theorem | | ⊢ |
| proveit.numbers.negation.complex_closure |
208 | theorem | | ⊢ |
| proveit.numbers.negation.real_closure |
209 | theorem | | ⊢ |
| proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero |
210 | instantiation | 257, 216, 217 | ⊢ |
| : , : , : |
211 | theorem | | ⊢ |
| proveit.numbers.division.div_complex_closure |
212 | instantiation | 218, 230, 219 | ⊢ |
| : , : |
213 | instantiation | 220, 221, 222 | ⊢ |
| : , : , : |
214 | instantiation | 257, 239, 223 | ⊢ |
| : , : , : |
215 | instantiation | 257, 252, 224 | ⊢ |
| : , : , : |
216 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero |
217 | instantiation | 257, 225, 226 | ⊢ |
| : , : , : |
218 | theorem | | ⊢ |
| proveit.numbers.exponentiation.exp_complex_closure |
219 | instantiation | 257, 239, 227 | ⊢ |
| : , : , : |
220 | theorem | | ⊢ |
| proveit.logic.equality.sub_left_side_into |
221 | instantiation | 228, 237 | ⊢ |
| : |
222 | instantiation | 229, 230 | ⊢ |
| : |
223 | instantiation | 231, 232, 233 | ⊢ |
| : , : , : |
224 | instantiation | 257, 234, 235 | ⊢ |
| : , : , : |
225 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero |
226 | instantiation | 257, 236, 237 | ⊢ |
| : , : , : |
227 | instantiation | 257, 246, 238 | ⊢ |
| : , : , : |
228 | theorem | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos |
229 | theorem | | ⊢ |
| proveit.numbers.exponentiation.complex_x_to_first_power_is_x |
230 | instantiation | 257, 239, 240 | ⊢ |
| : , : , : |
231 | theorem | | ⊢ |
| proveit.logic.sets.inclusion.unfold_subset_eq |
232 | instantiation | 241, 242 | ⊢ |
| : , : |
233 | axiom | | ⊢ |
| proveit.physics.quantum.QPE._t_in_natural_pos |
234 | instantiation | 243, 245, 244 | ⊢ |
| : , : |
235 | assumption | | ⊢ |
236 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int |
237 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.posnat2 |
238 | instantiation | 257, 252, 245 | ⊢ |
| : , : , : |
239 | theorem | | ⊢ |
| proveit.numbers.number_sets.complex_numbers.real_within_complex |
240 | instantiation | 257, 246, 247 | ⊢ |
| : , : , : |
241 | theorem | | ⊢ |
| proveit.logic.sets.inclusion.relax_proper_subset |
242 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.nat_pos_within_real |
243 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.int_interval_within_int |
244 | instantiation | 248, 249, 250 | ⊢ |
| : , : |
245 | instantiation | 257, 258, 251 | ⊢ |
| : , : , : |
246 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.rational_within_real |
247 | instantiation | 257, 252, 256 | ⊢ |
| : , : , : |
248 | theorem | | ⊢ |
| proveit.numbers.addition.add_int_closure_bin |
249 | instantiation | 257, 253, 254 | ⊢ |
| : , : , : |
250 | instantiation | 255, 256 | ⊢ |
| : |
251 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat1 |
252 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.int_within_rational |
253 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.nat_pos_within_int |
254 | theorem | | ⊢ |
| proveit.physics.quantum.QPE._two_pow_t_minus_one_is_nat_pos |
255 | theorem | | ⊢ |
| proveit.numbers.negation.int_closure |
256 | instantiation | 257, 258, 259 | ⊢ |
| : , : , : |
257 | theorem | | ⊢ |
| proveit.logic.sets.inclusion.superset_membership_from_proper_subset |
258 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.nat_within_int |
259 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat2 |
*equality replacement requirements |