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