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