| step type | requirements | statement |
0 | instantiation | 1, 2, 3 | ⊢ |
| : , : , : |
1 | reference | 26 | ⊢ |
2 | instantiation | 4, 5, 6, 7, 43 | ⊢ |
| : , : , : |
3 | instantiation | 128, 8, 9 | ⊢ |
| : , : , : |
4 | theorem | | ⊢ |
| proveit.numbers.division.strong_div_from_denom_bound__all_pos |
5 | instantiation | 236, 11, 10 | ⊢ |
| : , : , : |
6 | instantiation | 236, 11, 12 | ⊢ |
| : , : , : |
7 | instantiation | 13, 14, 15 | ⊢ |
| : |
8 | instantiation | 16, 48, 76, 17, 42, 18* | ⊢ |
| : , : , : |
9 | instantiation | 19, 55, 20, 204, 21, 22* | ⊢ |
| : , : , : |
10 | instantiation | 236, 24, 23 | ⊢ |
| : , : , : |
11 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos |
12 | instantiation | 236, 24, 53 | ⊢ |
| : , : , : |
13 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.pos_real_is_real_pos |
14 | instantiation | 25, 56, 177 | ⊢ |
| : , : |
15 | instantiation | 26, 27 | ⊢ |
| : , : , : |
16 | theorem | | ⊢ |
| proveit.numbers.division.strong_div_from_numer_bound__pos_denom |
17 | instantiation | 28, 76, 140, 29, 30, 31*, 32* | ⊢ |
| : , : , : |
18 | instantiation | 33, 34, 35 | ⊢ |
| : |
19 | theorem | | ⊢ |
| proveit.numbers.exponentiation.exp_eq_real |
20 | instantiation | 236, 36, 37 | ⊢ |
| : , : , : |
21 | instantiation | 38, 50, 189, 66, 39* | ⊢ |
| : , : |
22 | instantiation | 40, 41 | ⊢ |
| : |
23 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.posnat4 |
24 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos |
25 | theorem | | ⊢ |
| proveit.numbers.exponentiation.exp_real_closure_nat_power |
26 | axiom | | ⊢ |
| proveit.numbers.ordering.transitivity_less_less |
27 | instantiation | 113, 42, 43 | ⊢ |
| : , : |
28 | theorem | | ⊢ |
| proveit.numbers.addition.strong_bound_via_left_term_bound |
29 | instantiation | 236, 215, 44 | ⊢ |
| : , : , : |
30 | instantiation | 142, 45 | ⊢ |
| : |
31 | instantiation | 172, 46, 47 | ⊢ |
| : , : , : |
32 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.add_5_4 |
33 | theorem | | ⊢ |
| proveit.numbers.division.frac_cancel_complete |
34 | instantiation | 236, 226, 48 | ⊢ |
| : , : , : |
35 | instantiation | 233, 53 | ⊢ |
| : |
36 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.real_nonneg_within_real |
37 | instantiation | 49, 50 | ⊢ |
| : |
38 | theorem | | ⊢ |
| proveit.numbers.absolute_value.abs_eq |
39 | instantiation | 51, 52 | ⊢ |
| : |
40 | theorem | | ⊢ |
| proveit.numbers.exponentiation.exponentiated_one |
41 | instantiation | 236, 226, 55 | ⊢ |
| : , : , : |
42 | instantiation | 142, 53 | ⊢ |
| : |
43 | instantiation | 54, 55, 96, 56, 57, 58, 59* | ⊢ |
| : , : , : |
44 | instantiation | 236, 231, 60 | ⊢ |
| : , : , : |
45 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.posnat5 |
46 | instantiation | 61, 63 | ⊢ |
| : |
47 | instantiation | 62, 63, 64 | ⊢ |
| : , : |
48 | instantiation | 236, 215, 65 | ⊢ |
| : , : , : |
49 | theorem | | ⊢ |
| proveit.numbers.absolute_value.abs_complex_closure |
50 | instantiation | 128, 189, 66 | ⊢ |
| : , : , : |
51 | theorem | | ⊢ |
| proveit.numbers.absolute_value.abs_non_neg_elim |
52 | instantiation | 81, 229 | ⊢ |
| : |
53 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.posnat9 |
54 | theorem | | ⊢ |
| proveit.numbers.exponentiation.exp_pos_less |
55 | instantiation | 236, 215, 67 | ⊢ |
| : , : , : |
56 | instantiation | 236, 68, 69 | ⊢ |
| : , : , : |
57 | instantiation | 113, 70, 71 | ⊢ |
| : , : |
58 | instantiation | 142, 85 | ⊢ |
| : |
59 | instantiation | 155, 72, 73, 74 | ⊢ |
| : , : , : , : |
60 | instantiation | 236, 228, 75 | ⊢ |
| : , : , : |
61 | theorem | | ⊢ |
| proveit.numbers.addition.elim_zero_right |
62 | theorem | | ⊢ |
| proveit.numbers.addition.commutation |
63 | instantiation | 236, 226, 76 | ⊢ |
| : , : , : |
64 | theorem | | ⊢ |
| proveit.numbers.number_sets.complex_numbers.zero_is_complex |
65 | instantiation | 236, 231, 77 | ⊢ |
| : , : , : |
66 | instantiation | 128, 78, 79 | ⊢ |
| : , : , : |
67 | instantiation | 236, 231, 80 | ⊢ |
| : , : , : |
68 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.real_pos_within_real |
69 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.pi_is_real_pos |
70 | instantiation | 81, 116 | ⊢ |
| : |
71 | instantiation | 82, 83 | ⊢ |
| : , : |
72 | instantiation | 84, 85, 86 | ⊢ |
| : , : |
73 | instantiation | 87, 177, 88, 89, 90 | ⊢ |
| : , : , : , : |
74 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.mult_3_3 |
75 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat5 |
76 | instantiation | 236, 215, 91 | ⊢ |
| : , : , : |
77 | instantiation | 236, 228, 92 | ⊢ |
| : , : , : |
78 | instantiation | 172, 93, 129 | ⊢ |
| : , : , : |
79 | instantiation | 94, 238, 95 | ⊢ |
| : , : |
80 | instantiation | 236, 228, 177 | ⊢ |
| : , : , : |
81 | theorem | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.natural_lower_bound |
82 | theorem | | ⊢ |
| proveit.logic.booleans.conjunction.left_from_and |
83 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.pi_between_3_and_4 |
84 | theorem | | ⊢ |
| proveit.numbers.exponentiation.exp_nat_pos_expansion |
85 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.posnat2 |
86 | instantiation | 236, 226, 96 | ⊢ |
| : , : , : |
87 | theorem | | ⊢ |
| proveit.core_expr_types.operations.operands_substitution_via_tuple |
88 | instantiation | 97, 177 | ⊢ |
| : , : |
89 | instantiation | 192 | ⊢ |
| : , : |
90 | instantiation | 98 | ⊢ |
| : |
91 | instantiation | 236, 231, 99 | ⊢ |
| : , : , : |
92 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat9 |
93 | modus ponens | 100, 101 | ⊢ |
94 | theorem | | ⊢ |
| proveit.numbers.modular.int_mod_elimination |
95 | instantiation | 105, 106, 107, 230, 102 | ⊢ |
| : , : , : |
96 | instantiation | 236, 215, 103 | ⊢ |
| : , : , : |
97 | theorem | | ⊢ |
| proveit.core_expr_types.tuples.range_from1_len_typical_eq |
98 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.reduce_2_repeats |
99 | instantiation | 236, 228, 104 | ⊢ |
| : , : , : |
100 | theorem | | ⊢ |
| proveit.physics.quantum.QPE._alpha_ideal_case |
101 | instantiation | 105, 106, 107, 122, 108 | ⊢ |
| : , : , : |
102 | instantiation | 113, 109, 110 | ⊢ |
| : , : |
103 | instantiation | 236, 231, 111 | ⊢ |
| : , : , : |
104 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat4 |
105 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.in_interval |
106 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.zero_is_int |
107 | instantiation | 112, 232, 148 | ⊢ |
| : , : |
108 | instantiation | 113, 114, 115 | ⊢ |
| : , : |
109 | instantiation | 172, 114, 129 | ⊢ |
| : , : , : |
110 | instantiation | 172, 115, 129 | ⊢ |
| : , : , : |
111 | instantiation | 236, 228, 116 | ⊢ |
| : , : , : |
112 | theorem | | ⊢ |
| proveit.numbers.addition.add_int_closure_bin |
113 | theorem | | ⊢ |
| proveit.logic.booleans.conjunction.and_if_both |
114 | instantiation | 117, 227, 140, 195, 118, 119, 120* | ⊢ |
| : , : , : |
115 | instantiation | 121, 122, 232, 148, 123, 124 | ⊢ |
| : , : , : |
116 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat3 |
117 | theorem | | ⊢ |
| proveit.numbers.multiplication.weak_bound_via_right_factor_bound |
118 | instantiation | 125, 140, 204, 141 | ⊢ |
| : , : , : |
119 | instantiation | 126, 132 | ⊢ |
| : , : |
120 | instantiation | 127, 220 | ⊢ |
| : |
121 | theorem | | ⊢ |
| proveit.numbers.ordering.less_add_right_weak_int |
122 | instantiation | 128, 230, 129 | ⊢ |
| : , : , : |
123 | instantiation | 130, 227, 195, 204, 131, 132, 203* | ⊢ |
| : , : , : |
124 | instantiation | 133, 134, 135 | ⊢ |
| : , : |
125 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.interval_co_lower_bound |
126 | theorem | | ⊢ |
| proveit.numbers.ordering.relax_less |
127 | theorem | | ⊢ |
| proveit.numbers.multiplication.mult_zero_right |
128 | theorem | | ⊢ |
| proveit.logic.equality.sub_left_side_into |
129 | instantiation | 136, 227, 195, 206, 137, 138* | ⊢ |
| : , : , : |
130 | theorem | | ⊢ |
| proveit.numbers.multiplication.strong_bound_via_right_factor_bound |
131 | instantiation | 139, 140, 204, 141 | ⊢ |
| : , : , : |
132 | instantiation | 142, 238 | ⊢ |
| : |
133 | theorem | | ⊢ |
| proveit.numbers.ordering.relax_equal_to_less_eq |
134 | instantiation | 236, 215, 143 | ⊢ |
| : , : , : |
135 | instantiation | 196 | ⊢ |
| : |
136 | theorem | | ⊢ |
| proveit.numbers.multiplication.left_mult_eq_real |
137 | instantiation | 144, 145 | ⊢ |
| : , : |
138 | instantiation | 172, 146, 147 | ⊢ |
| : , : , : |
139 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.interval_co_upper_bound |
140 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.zero_is_real |
141 | axiom | | ⊢ |
| proveit.physics.quantum.QPE._phase_in_interval |
142 | theorem | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos |
143 | instantiation | 236, 231, 148 | ⊢ |
| : , : , : |
144 | theorem | | ⊢ |
| proveit.logic.equality.equals_reversal |
145 | instantiation | 149, 216, 160, 150, 161, 151*, 152* | ⊢ |
| : , : , : |
146 | instantiation | 172, 153, 154 | ⊢ |
| : , : , : |
147 | instantiation | 155, 156, 157, 158 | ⊢ |
| : , : , : , : |
148 | instantiation | 159, 221 | ⊢ |
| : |
149 | theorem | | ⊢ |
| proveit.numbers.addition.right_add_eq_rational |
150 | instantiation | 172, 160, 161 | ⊢ |
| : , : , : |
151 | instantiation | 162, 194 | ⊢ |
| : |
152 | instantiation | 200, 163, 164 | ⊢ |
| : , : , : |
153 | instantiation | 165, 189, 190, 166, 167 | ⊢ |
| : , : , : , : , : |
154 | instantiation | 200, 168, 169 | ⊢ |
| : , : , : |
155 | theorem | | ⊢ |
| proveit.logic.equality.four_chain_transitivity |
156 | instantiation | 210, 170 | ⊢ |
| : , : , : |
157 | instantiation | 210, 171 | ⊢ |
| : , : , : |
158 | instantiation | 219, 190 | ⊢ |
| : |
159 | theorem | | ⊢ |
| proveit.numbers.negation.int_closure |
160 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.zero_is_rational |
161 | instantiation | 172, 173, 174 | ⊢ |
| : , : , : |
162 | theorem | | ⊢ |
| proveit.numbers.addition.elim_zero_left |
163 | instantiation | 175, 176, 177, 229, 178, 179, 182, 180, 194 | ⊢ |
| : , : , : , : , : , : |
164 | instantiation | 181, 194, 182, 183 | ⊢ |
| : , : , : |
165 | theorem | | ⊢ |
| proveit.numbers.division.mult_frac_cancel_numer_left |
166 | instantiation | 236, 185, 184 | ⊢ |
| : , : , : |
167 | instantiation | 236, 185, 186 | ⊢ |
| : , : , : |
168 | instantiation | 210, 187 | ⊢ |
| : , : , : |
169 | instantiation | 210, 188 | ⊢ |
| : , : , : |
170 | instantiation | 212, 189 | ⊢ |
| : |
171 | instantiation | 212, 190 | ⊢ |
| : |
172 | theorem | | ⊢ |
| proveit.logic.equality.sub_right_side_into |
173 | instantiation | 191, 230 | ⊢ |
| : |
174 | assumption | | ⊢ |
175 | theorem | | ⊢ |
| proveit.numbers.addition.disassociation |
176 | axiom | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.zero_in_nats |
177 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat2 |
178 | theorem | | ⊢ |
| proveit.core_expr_types.tuples.tuple_len_0_typical_eq |
179 | instantiation | 192 | ⊢ |
| : , : |
180 | instantiation | 193, 194 | ⊢ |
| : |
181 | theorem | | ⊢ |
| proveit.numbers.addition.subtraction.add_cancel_triple_32 |
182 | instantiation | 236, 226, 195 | ⊢ |
| : , : , : |
183 | instantiation | 196 | ⊢ |
| : |
184 | instantiation | 236, 198, 197 | ⊢ |
| : , : , : |
185 | theorem | | ⊢ |
| proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero |
186 | instantiation | 236, 198, 199 | ⊢ |
| : , : , : |
187 | instantiation | 200, 201, 202 | ⊢ |
| : , : , : |
188 | instantiation | 210, 203 | ⊢ |
| : , : , : |
189 | instantiation | 236, 226, 204 | ⊢ |
| : , : , : |
190 | instantiation | 236, 226, 205 | ⊢ |
| : , : , : |
191 | axiom | | ⊢ |
| proveit.physics.quantum.QPE._delta_b_def |
192 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.tuple_len_2_typical_eq |
193 | theorem | | ⊢ |
| proveit.numbers.negation.complex_closure |
194 | instantiation | 236, 226, 206 | ⊢ |
| : , : , : |
195 | theorem | | ⊢ |
| proveit.physics.quantum.QPE._phase_is_real |
196 | axiom | | ⊢ |
| proveit.logic.equality.equals_reflexivity |
197 | instantiation | 236, 208, 207 | ⊢ |
| : , : , : |
198 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero |
199 | instantiation | 236, 208, 209 | ⊢ |
| : , : , : |
200 | axiom | | ⊢ |
| proveit.logic.equality.equals_transitivity |
201 | instantiation | 210, 211 | ⊢ |
| : , : , : |
202 | instantiation | 212, 220 | ⊢ |
| : |
203 | instantiation | 213, 220 | ⊢ |
| : |
204 | instantiation | 236, 215, 214 | ⊢ |
| : , : , : |
205 | instantiation | 236, 215, 223 | ⊢ |
| : , : , : |
206 | instantiation | 236, 215, 216 | ⊢ |
| : , : , : |
207 | instantiation | 236, 217, 238 | ⊢ |
| : , : , : |
208 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero |
209 | instantiation | 236, 217, 218 | ⊢ |
| : , : , : |
210 | axiom | | ⊢ |
| proveit.logic.equality.substitution |
211 | instantiation | 219, 220 | ⊢ |
| : |
212 | theorem | | ⊢ |
| proveit.numbers.division.frac_one_denom |
213 | theorem | | ⊢ |
| proveit.numbers.multiplication.elim_one_right |
214 | instantiation | 236, 231, 221 | ⊢ |
| : , : , : |
215 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.rational_within_real |
216 | instantiation | 222, 223, 224, 225 | ⊢ |
| : , : |
217 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int |
218 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.posnat1 |
219 | theorem | | ⊢ |
| proveit.numbers.multiplication.elim_one_left |
220 | instantiation | 236, 226, 227 | ⊢ |
| : , : , : |
221 | instantiation | 236, 228, 229 | ⊢ |
| : , : , : |
222 | theorem | | ⊢ |
| proveit.numbers.division.div_rational_closure |
223 | instantiation | 236, 231, 230 | ⊢ |
| : , : , : |
224 | instantiation | 236, 231, 232 | ⊢ |
| : , : , : |
225 | instantiation | 233, 238 | ⊢ |
| : |
226 | theorem | | ⊢ |
| proveit.numbers.number_sets.complex_numbers.real_within_complex |
227 | instantiation | 234, 235, 238 | ⊢ |
| : , : , : |
228 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.nat_within_int |
229 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat1 |
230 | theorem | | ⊢ |
| proveit.physics.quantum.QPE._best_round_is_int |
231 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.int_within_rational |
232 | instantiation | 236, 237, 238 | ⊢ |
| : , : , : |
233 | theorem | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos |
234 | theorem | | ⊢ |
| proveit.logic.sets.inclusion.unfold_subset_eq |
235 | instantiation | 239, 240 | ⊢ |
| : , : |
236 | theorem | | ⊢ |
| proveit.logic.sets.inclusion.superset_membership_from_proper_subset |
237 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.nat_pos_within_int |
238 | theorem | | ⊢ |
| proveit.physics.quantum.QPE._two_pow_t_is_nat_pos |
239 | theorem | | ⊢ |
| proveit.logic.sets.inclusion.relax_proper_subset |
240 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.nat_pos_within_real |
*equality replacement requirements |