| step type | requirements | statement |
0 | instantiation | 1, 2, 3, 4 | ⊢ |
| : , : , : |
1 | theorem | | ⊢ |
| proveit.numbers.division.frac_cancel_left |
2 | instantiation | 230, 5, 6 | ⊢ |
| : , : , : |
3 | instantiation | 190, 7, 8 | ⊢ |
| : |
4 | instantiation | 230, 222, 9 | ⊢ |
| : , : , : |
5 | theorem | | ⊢ |
| proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero |
6 | instantiation | 230, 10, 11 | ⊢ |
| : , : , : |
7 | instantiation | 230, 222, 19 | ⊢ |
| : , : , : |
8 | instantiation | 12, 13 | ⊢ |
| : , : |
9 | instantiation | 14, 15 | ⊢ |
| : |
10 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero |
11 | instantiation | 230, 16, 17 | ⊢ |
| : , : , : |
12 | theorem | | ⊢ |
| proveit.logic.equality.not_equals_symmetry |
13 | instantiation | 18, 51, 19, 20 | ⊢ |
| : , : |
14 | theorem | | ⊢ |
| proveit.trigonometry.real_closure |
15 | instantiation | 79, 21, 22 | ⊢ |
| : , : , : |
16 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero |
17 | instantiation | 230, 23, 212 | ⊢ |
| : , : , : |
18 | theorem | | ⊢ |
| proveit.numbers.ordering.less_is_not_eq |
19 | instantiation | 24, 51, 194, 26 | ⊢ |
| : , : , : |
20 | instantiation | 25, 51, 194, 26 | ⊢ |
| : , : , : |
21 | instantiation | 107, 27, 120 | ⊢ |
| : , : |
22 | instantiation | 83, 138, 232, 225, 139, 28, 143, 96, 97 | ⊢ |
| : , : , : , : , : , : |
23 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int |
24 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real |
25 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound |
26 | instantiation | 29, 30 | ⊢ |
| : |
27 | instantiation | 107, 170, 119 | ⊢ |
| : , : |
28 | instantiation | 213 | ⊢ |
| : , : |
29 | theorem | | ⊢ |
| proveit.trigonometry.sine_pos_interval |
30 | instantiation | 31, 51, 119, 32, 33 | ⊢ |
| : , : , : |
31 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.in_IntervalOO |
32 | instantiation | 230, 146, 37 | ⊢ |
| : , : , : |
33 | instantiation | 34, 35, 36 | ⊢ |
| : , : |
34 | theorem | | ⊢ |
| proveit.logic.booleans.conjunction.and_if_both |
35 | instantiation | 106, 37 | ⊢ |
| : |
36 | instantiation | 38, 39, 40 | ⊢ |
| : , : , : |
37 | instantiation | 41, 145, 147 | ⊢ |
| : , : |
38 | axiom | | ⊢ |
| proveit.numbers.ordering.transitivity_less_less |
39 | instantiation | 79, 42, 60 | ⊢ |
| : , : , : |
40 | instantiation | 90, 88, 43, 44, 45*, 46* | ⊢ |
| : , : , : |
41 | theorem | | ⊢ |
| proveit.numbers.multiplication.mult_real_pos_closure_bin |
42 | instantiation | 47, 48, 49 | ⊢ |
| : , : , : |
43 | instantiation | 107, 108, 51 | ⊢ |
| : , : |
44 | instantiation | 50, 108, 51, 119, 82, 52 | ⊢ |
| : , : , : |
45 | instantiation | 180, 53, 54 | ⊢ |
| : , : , : |
46 | instantiation | 55, 56, 57* | ⊢ |
| : , : |
47 | theorem | | ⊢ |
| proveit.numbers.ordering.transitivity_less_less_eq |
48 | instantiation | 58, 59, 60 | ⊢ |
| : , : , : |
49 | instantiation | 61, 119, 62, 175, 63, 64, 65*, 66* | ⊢ |
| : , : , : |
50 | theorem | | ⊢ |
| proveit.numbers.multiplication.strong_bound_via_right_factor_bound |
51 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.zero_is_real |
52 | instantiation | 67, 157 | ⊢ |
| : |
53 | instantiation | 121, 68 | ⊢ |
| : , : , : |
54 | instantiation | 69, 70 | ⊢ |
| : |
55 | theorem | | ⊢ |
| proveit.logic.equality.equals_reversal |
56 | instantiation | 71, 138, 232, 225, 139, 72, 87, 96 | ⊢ |
| : , : , : , : , : , : |
57 | instantiation | 180, 73, 74 | ⊢ |
| : , : , : |
58 | theorem | | ⊢ |
| proveit.logic.equality.sub_left_side_into |
59 | instantiation | 75, 225, 145, 147, 76, 77* | ⊢ |
| : , : , : , : , : , : |
60 | instantiation | 121, 78 | ⊢ |
| : , : , : |
61 | theorem | | ⊢ |
| proveit.numbers.multiplication.weak_bound_via_right_factor_bound |
62 | instantiation | 107, 170, 120 | ⊢ |
| : , : |
63 | instantiation | 79, 80, 81 | ⊢ |
| : , : , : |
64 | instantiation | 154, 82 | ⊢ |
| : , : |
65 | instantiation | 83, 225, 232, 138, 84, 139, 96, 143, 97 | ⊢ |
| : , : , : , : , : , : |
66 | instantiation | 85, 96, 87 | ⊢ |
| : , : |
67 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos |
68 | instantiation | 86, 87 | ⊢ |
| : |
69 | theorem | | ⊢ |
| proveit.numbers.addition.elim_zero_left |
70 | instantiation | 230, 222, 88 | ⊢ |
| : , : , : |
71 | theorem | | ⊢ |
| proveit.numbers.multiplication.distribute_through_sum |
72 | instantiation | 213 | ⊢ |
| : , : |
73 | instantiation | 121, 89 | ⊢ |
| : , : , : |
74 | instantiation | 214, 96 | ⊢ |
| : |
75 | theorem | | ⊢ |
| proveit.numbers.multiplication.strong_bound_via_factor_bound |
76 | instantiation | 90, 169, 223, 91, 92, 93*, 94* | ⊢ |
| : , : , : |
77 | instantiation | 95, 225, 96, 97 | ⊢ |
| : , : , : , : |
78 | instantiation | 180, 98, 99 | ⊢ |
| : , : , : |
79 | theorem | | ⊢ |
| proveit.logic.equality.sub_right_side_into |
80 | instantiation | 100, 101, 102 | ⊢ |
| : , : |
81 | instantiation | 103, 212, 104, 143, 191, 105* | ⊢ |
| : , : |
82 | instantiation | 106, 145 | ⊢ |
| : |
83 | theorem | | ⊢ |
| proveit.numbers.multiplication.disassociation |
84 | instantiation | 213 | ⊢ |
| : , : |
85 | theorem | | ⊢ |
| proveit.numbers.multiplication.commutation |
86 | theorem | | ⊢ |
| proveit.numbers.multiplication.mult_zero_right |
87 | instantiation | 230, 222, 175 | ⊢ |
| : , : , : |
88 | instantiation | 107, 108, 119 | ⊢ |
| : , : |
89 | instantiation | 109, 221, 229, 110* | ⊢ |
| : , : , : , : |
90 | theorem | | ⊢ |
| proveit.numbers.addition.strong_bound_via_left_term_bound |
91 | instantiation | 230, 226, 111 | ⊢ |
| : , : , : |
92 | instantiation | 112, 223, 170, 194, 113, 114 | ⊢ |
| : , : , : |
93 | instantiation | 115, 149, 216, 116 | ⊢ |
| : , : , : |
94 | instantiation | 180, 117, 118 | ⊢ |
| : , : , : |
95 | theorem | | ⊢ |
| proveit.numbers.multiplication.elim_one_any |
96 | instantiation | 230, 222, 119 | ⊢ |
| : , : , : |
97 | instantiation | 230, 222, 120 | ⊢ |
| : , : , : |
98 | instantiation | 121, 122 | ⊢ |
| : , : , : |
99 | instantiation | 123, 191 | ⊢ |
| : |
100 | theorem | | ⊢ |
| proveit.numbers.absolute_value.weak_upper_bound |
101 | instantiation | 124, 152, 175, 153 | ⊢ |
| : , : , : |
102 | instantiation | 154, 125 | ⊢ |
| : , : |
103 | theorem | | ⊢ |
| proveit.numbers.absolute_value.abs_prod |
104 | instantiation | 213 | ⊢ |
| : , : |
105 | instantiation | 126, 127 | ⊢ |
| : |
106 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.positive_if_in_real_pos |
107 | theorem | | ⊢ |
| proveit.numbers.multiplication.mult_real_closure_bin |
108 | instantiation | 230, 226, 128 | ⊢ |
| : , : , : |
109 | theorem | | ⊢ |
| proveit.numbers.addition.rational_pair_addition |
110 | instantiation | 180, 129, 130 | ⊢ |
| : , : , : |
111 | instantiation | 230, 228, 131 | ⊢ |
| : , : , : |
112 | theorem | | ⊢ |
| proveit.numbers.ordering.less_eq_add_right_strong |
113 | instantiation | 132, 223, 194, 133, 134, 135, 136* | ⊢ |
| : , : , : |
114 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.less_0_1 |
115 | theorem | | ⊢ |
| proveit.numbers.addition.subtraction.subtract_from_add |
116 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.add_1_1 |
117 | instantiation | 137, 138, 232, 225, 139, 140, 143, 149, 141 | ⊢ |
| : , : , : , : , : , : |
118 | instantiation | 142, 149, 143, 144 | ⊢ |
| : , : , : |
119 | instantiation | 230, 146, 145 | ⊢ |
| : , : , : |
120 | instantiation | 230, 146, 147 | ⊢ |
| : , : , : |
121 | axiom | | ⊢ |
| proveit.logic.equality.substitution |
122 | instantiation | 148, 149, 191, 150* | ⊢ |
| : , : |
123 | theorem | | ⊢ |
| proveit.numbers.absolute_value.abs_even |
124 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.interval_co_lower_bound |
125 | instantiation | 151, 152, 175, 153 | ⊢ |
| : , : , : |
126 | theorem | | ⊢ |
| proveit.numbers.absolute_value.abs_non_neg_elim |
127 | instantiation | 154, 155 | ⊢ |
| : , : |
128 | instantiation | 230, 156, 157 | ⊢ |
| : , : , : |
129 | instantiation | 198, 232, 158, 159, 160, 161 | ⊢ |
| : , : , : , : |
130 | instantiation | 162, 163, 164 | ⊢ |
| : |
131 | instantiation | 230, 231, 165 | ⊢ |
| : , : , : |
132 | theorem | | ⊢ |
| proveit.numbers.exponentiation.exp_monotonicity_large_base_less_eq |
133 | instantiation | 188, 189, 167 | ⊢ |
| : , : , : |
134 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.less_1_2 |
135 | instantiation | 166, 167 | ⊢ |
| : |
136 | instantiation | 168, 216 | ⊢ |
| : |
137 | theorem | | ⊢ |
| proveit.numbers.addition.disassociation |
138 | axiom | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.zero_in_nats |
139 | theorem | | ⊢ |
| proveit.core_expr_types.tuples.tuple_len_0_typical_eq |
140 | instantiation | 213 | ⊢ |
| : , : |
141 | instantiation | 230, 222, 169 | ⊢ |
| : , : , : |
142 | theorem | | ⊢ |
| proveit.numbers.addition.subtraction.add_cancel_triple_23 |
143 | instantiation | 230, 222, 170 | ⊢ |
| : , : , : |
144 | instantiation | 171 | ⊢ |
| : |
145 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.pi_is_real_pos |
146 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.real_pos_within_real |
147 | instantiation | 172, 173 | ⊢ |
| : |
148 | theorem | | ⊢ |
| proveit.numbers.multiplication.mult_neg_left |
149 | instantiation | 230, 222, 194 | ⊢ |
| : , : , : |
150 | instantiation | 214, 191 | ⊢ |
| : |
151 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.interval_co_upper_bound |
152 | instantiation | 174, 175 | ⊢ |
| : |
153 | theorem | | ⊢ |
| proveit.physics.quantum.QPE._scaled_delta_b_round_in_interval |
154 | theorem | | ⊢ |
| proveit.numbers.ordering.relax_less |
155 | instantiation | 176, 205 | ⊢ |
| : |
156 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational |
157 | instantiation | 177, 178, 179 | ⊢ |
| : , : |
158 | instantiation | 213 | ⊢ |
| : , : |
159 | instantiation | 213 | ⊢ |
| : , : |
160 | instantiation | 180, 181, 182 | ⊢ |
| : , : , : |
161 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.mult_2_2 |
162 | theorem | | ⊢ |
| proveit.numbers.division.frac_cancel_complete |
163 | instantiation | 230, 222, 183 | ⊢ |
| : , : , : |
164 | instantiation | 211, 184 | ⊢ |
| : |
165 | instantiation | 185, 186, 225 | ⊢ |
| : , : |
166 | theorem | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.natural_pos_lower_bound |
167 | axiom | | ⊢ |
| proveit.physics.quantum.QPE._t_in_natural_pos |
168 | theorem | | ⊢ |
| proveit.numbers.exponentiation.complex_x_to_first_power_is_x |
169 | instantiation | 230, 226, 187 | ⊢ |
| : , : , : |
170 | instantiation | 188, 189, 205 | ⊢ |
| : , : , : |
171 | axiom | | ⊢ |
| proveit.logic.equality.equals_reflexivity |
172 | theorem | | ⊢ |
| proveit.numbers.absolute_value.abs_nonzero_closure |
173 | instantiation | 190, 191, 192 | ⊢ |
| : |
174 | theorem | | ⊢ |
| proveit.numbers.negation.real_closure |
175 | instantiation | 193, 194, 223, 195 | ⊢ |
| : , : |
176 | theorem | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos |
177 | theorem | | ⊢ |
| proveit.numbers.division.div_rational_pos_closure |
178 | instantiation | 230, 197, 196 | ⊢ |
| : , : , : |
179 | instantiation | 230, 197, 212 | ⊢ |
| : , : , : |
180 | axiom | | ⊢ |
| proveit.logic.equality.equals_transitivity |
181 | instantiation | 198, 232, 199, 200, 201, 202 | ⊢ |
| : , : , : , : |
182 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.add_2_2 |
183 | instantiation | 230, 226, 203 | ⊢ |
| : , : , : |
184 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.posnat4 |
185 | theorem | | ⊢ |
| proveit.numbers.addition.add_nat_closure_bin |
186 | instantiation | 230, 204, 205 | ⊢ |
| : , : , : |
187 | instantiation | 230, 228, 206 | ⊢ |
| : , : , : |
188 | theorem | | ⊢ |
| proveit.logic.sets.inclusion.unfold_subset_eq |
189 | instantiation | 207, 208 | ⊢ |
| : , : |
190 | theorem | | ⊢ |
| proveit.numbers.number_sets.complex_numbers.nonzero_complex_is_complex_nonzero |
191 | instantiation | 230, 222, 209 | ⊢ |
| : , : , : |
192 | assumption | | ⊢ |
193 | theorem | | ⊢ |
| proveit.numbers.division.div_real_closure |
194 | instantiation | 230, 226, 210 | ⊢ |
| : , : , : |
195 | instantiation | 211, 212 | ⊢ |
| : |
196 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.posnat1 |
197 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos |
198 | axiom | | ⊢ |
| proveit.core_expr_types.operations.operands_substitution |
199 | instantiation | 213 | ⊢ |
| : , : |
200 | instantiation | 213 | ⊢ |
| : , : |
201 | instantiation | 214, 216 | ⊢ |
| : |
202 | instantiation | 215, 216 | ⊢ |
| : |
203 | instantiation | 230, 228, 217 | ⊢ |
| : , : , : |
204 | theorem | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat |
205 | theorem | | ⊢ |
| proveit.physics.quantum.QPE._two_pow_t_is_nat_pos |
206 | instantiation | 218, 221 | ⊢ |
| : |
207 | theorem | | ⊢ |
| proveit.logic.sets.inclusion.relax_proper_subset |
208 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.nat_pos_within_real |
209 | instantiation | 219, 220 | ⊢ |
| : |
210 | instantiation | 230, 228, 221 | ⊢ |
| : , : , : |
211 | theorem | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos |
212 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.posnat2 |
213 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.tuple_len_2_typical_eq |
214 | theorem | | ⊢ |
| proveit.numbers.multiplication.elim_one_left |
215 | theorem | | ⊢ |
| proveit.numbers.multiplication.elim_one_right |
216 | instantiation | 230, 222, 223 | ⊢ |
| : , : , : |
217 | instantiation | 230, 231, 224 | ⊢ |
| : , : , : |
218 | theorem | | ⊢ |
| proveit.numbers.negation.int_closure |
219 | theorem | | ⊢ |
| proveit.physics.quantum.QPE._delta_b_is_real |
220 | theorem | | ⊢ |
| proveit.physics.quantum.QPE._best_round_is_int |
221 | instantiation | 230, 231, 225 | ⊢ |
| : , : , : |
222 | theorem | | ⊢ |
| proveit.numbers.number_sets.complex_numbers.real_within_complex |
223 | instantiation | 230, 226, 227 | ⊢ |
| : , : , : |
224 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat4 |
225 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat1 |
226 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.rational_within_real |
227 | instantiation | 230, 228, 229 | ⊢ |
| : , : , : |
228 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.int_within_rational |
229 | instantiation | 230, 231, 232 | ⊢ |
| : , : , : |
230 | theorem | | ⊢ |
| proveit.logic.sets.inclusion.superset_membership_from_proper_subset |
231 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.nat_within_int |
232 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat2 |
*equality replacement requirements |