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