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