| step type | requirements | statement |
0 | instantiation | 1, 2 | ⊢ |
| : , : |
1 | theorem | | ⊢ |
| proveit.logic.sets.membership.fold_not_in_set |
2 | instantiation | 3, 4, 5, 6 | ⊢ |
| : , : |
3 | theorem | | ⊢ |
| proveit.logic.booleans.implication.modus_tollens_denial |
4 | instantiation | 7 | ⊢ |
| : |
5 | deduction | 8 | ⊢ |
6 | theorem | | ⊢ |
| proveit.logic.booleans.negation.not_false |
7 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.rational_membership_is_bool |
8 | modus ponens | 9, 10 | ⊢ |
9 | instantiation | 11, 12, 13, 151 | ⊢ |
| : , : , : , : , : , : |
10 | instantiation | 23, 14, 15 | ⊢ |
| : , : |
11 | theorem | | ⊢ |
| proveit.logic.booleans.quantification.existence.skolem_elim |
12 | instantiation | 16 | ⊢ |
| : , : |
13 | instantiation | 16 | ⊢ |
| : , : |
14 | instantiation | 17, 94 | ⊢ |
| : |
15 | generalization | 18 | ⊢ |
16 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.tuple_len_2_typical_eq |
17 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.reduced_nat_pos_ratio |
18 | deduction | 19 | , , ⊢ |
19 | instantiation | 20, 21, 22 | , , , ⊢ |
| : |
20 | theorem | | ⊢ |
| proveit.logic.booleans.negation.negation_contradiction |
21 | instantiation | 23, 42, 24 | , , , ⊢ |
| : , : |
22 | instantiation | 25, 151, 26 | , , ⊢ |
| : |
23 | theorem | | ⊢ |
| proveit.logic.booleans.conjunction.and_if_both |
24 | instantiation | 43, 27, 75, 28 | , , , ⊢ |
| : , : |
25 | instantiation | 29, 131, 154, 30 | , , ⊢ |
| : , : |
26 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.less_1_2 |
27 | instantiation | 149, 54, 154 | ⊢ |
| : , : , : |
28 | instantiation | 31, 49, 32, 33, 51 | , , , ⊢ |
| : , : , : |
29 | theorem | | ⊢ |
| proveit.numbers.divisibility.GCD_one_def |
30 | instantiation | 34, 79 | ⊢ |
| : , : |
31 | theorem | | ⊢ |
| proveit.numbers.divisibility.common_factor_elimination |
32 | instantiation | 149, 144, 35 | ⊢ |
| : , : , : |
33 | instantiation | 86, 36, 37 | , , , ⊢ |
| : , : , : |
34 | theorem | | ⊢ |
| proveit.logic.booleans.conjunction.right_from_and |
35 | instantiation | 152, 153, 55 | ⊢ |
| : , : , : |
36 | instantiation | 38, 39, 47 | , , , ⊢ |
| : , : , : |
37 | instantiation | 40, 49 | ⊢ |
| : |
38 | theorem | | ⊢ |
| proveit.logic.equality.sub_left_side_into |
39 | instantiation | 41, 75, 44, 151, 42 | , , , ⊢ |
| : , : , : |
40 | theorem | | ⊢ |
| proveit.numbers.exponentiation.square_expansion |
41 | theorem | | ⊢ |
| proveit.numbers.divisibility.common_exponent_introduction |
42 | instantiation | 43, 44, 75, 45 | , , , ⊢ |
| : , : |
43 | theorem | | ⊢ |
| proveit.numbers.divisibility.even__if__power_is_even |
44 | instantiation | 149, 54, 131 | ⊢ |
| : , : , : |
45 | instantiation | 86, 46, 47 | , , , ⊢ |
| : , : , : |
46 | instantiation | 48, 49, 50, 51 | ⊢ |
| : , : |
47 | instantiation | 86, 52, 53 | , , , ⊢ |
| : , : , : |
48 | theorem | | ⊢ |
| proveit.numbers.divisibility.left_factor_divisibility |
49 | instantiation | 149, 144, 57 | ⊢ |
| : , : , : |
50 | instantiation | 149, 54, 55 | ⊢ |
| : , : , : |
51 | instantiation | 85, 151 | ⊢ |
| : |
52 | instantiation | 56, 57, 58, 122, 59 | , , , ⊢ |
| : , : , : |
53 | instantiation | 60, 61, 138, 151, 62* | , ⊢ |
| : , : , : |
54 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.nat_pos_within_int |
55 | instantiation | 63, 64, 96 | ⊢ |
| : , : |
56 | theorem | | ⊢ |
| proveit.numbers.exponentiation.exp_eq_real |
57 | instantiation | 149, 132, 65 | ⊢ |
| : , : , : |
58 | instantiation | 66, 71, 145 | , ⊢ |
| : , : |
59 | instantiation | 67, 145, 71, 68, 69, 70* | , , , ⊢ |
| : , : , : |
60 | theorem | | ⊢ |
| proveit.numbers.exponentiation.posnat_power_of_product |
61 | instantiation | 149, 144, 71 | ⊢ |
| : , : , : |
62 | instantiation | 72, 114, 73* | ⊢ |
| : |
63 | theorem | | ⊢ |
| proveit.numbers.exponentiation.exp_natpos_closure |
64 | instantiation | 149, 74, 154 | ⊢ |
| : , : , : |
65 | instantiation | 149, 139, 75 | ⊢ |
| : , : , : |
66 | theorem | | ⊢ |
| proveit.numbers.multiplication.mult_real_closure_bin |
67 | theorem | | ⊢ |
| proveit.numbers.multiplication.right_mult_eq_real |
68 | instantiation | 76, 122, 145, 77 | , ⊢ |
| : , : |
69 | instantiation | 78, 79 | ⊢ |
| : , : |
70 | instantiation | 86, 80, 81 | , ⊢ |
| : , : , : |
71 | instantiation | 149, 132, 82 | ⊢ |
| : , : , : |
72 | theorem | | ⊢ |
| proveit.numbers.exponentiation.square_abs_rational_simp |
73 | instantiation | 83, 151, 84 | ⊢ |
| : , : |
74 | theorem | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat |
75 | instantiation | 149, 146, 96 | ⊢ |
| : , : , : |
76 | theorem | | ⊢ |
| proveit.numbers.division.div_real_closure |
77 | instantiation | 85, 154 | ⊢ |
| : |
78 | theorem | | ⊢ |
| proveit.logic.booleans.conjunction.left_from_and |
79 | assumption | | ⊢ |
80 | instantiation | 86, 87, 88 | , ⊢ |
| : , : , : |
81 | instantiation | 89, 90, 91, 92 | ⊢ |
| : , : , : , : |
82 | instantiation | 149, 93, 94 | ⊢ |
| : , : , : |
83 | theorem | | ⊢ |
| proveit.numbers.exponentiation.nth_power_of_nth_root |
84 | instantiation | 95, 96 | ⊢ |
| : |
85 | theorem | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos |
86 | theorem | | ⊢ |
| proveit.logic.equality.sub_right_side_into |
87 | instantiation | 97, 111, 112, 98, 99 | , ⊢ |
| : , : , : , : , : |
88 | instantiation | 119, 100, 101 | ⊢ |
| : , : , : |
89 | theorem | | ⊢ |
| proveit.logic.equality.four_chain_transitivity |
90 | instantiation | 128, 102 | ⊢ |
| : , : , : |
91 | instantiation | 128, 103 | ⊢ |
| : , : , : |
92 | instantiation | 137, 111 | ⊢ |
| : |
93 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational |
94 | instantiation | 104, 105 | ⊢ |
| : |
95 | theorem | | ⊢ |
| proveit.numbers.number_sets.natural_numbers.natural_lower_bound |
96 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat2 |
97 | theorem | | ⊢ |
| proveit.numbers.division.mult_frac_cancel_denom_left |
98 | instantiation | 149, 107, 106 | ⊢ |
| : , : , : |
99 | instantiation | 149, 107, 108 | ⊢ |
| : , : , : |
100 | instantiation | 128, 109 | ⊢ |
| : , : , : |
101 | instantiation | 128, 110 | ⊢ |
| : , : , : |
102 | instantiation | 130, 111 | ⊢ |
| : |
103 | instantiation | 130, 112 | ⊢ |
| : |
104 | theorem | | ⊢ |
| proveit.numbers.absolute_value.abs_rational_nonzero_closure |
105 | instantiation | 113, 114, 115 | ⊢ |
| : |
106 | instantiation | 149, 116, 135 | ⊢ |
| : , : , : |
107 | theorem | | ⊢ |
| proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero |
108 | instantiation | 149, 116, 117 | ⊢ |
| : , : , : |
109 | instantiation | 128, 118 | ⊢ |
| : , : , : |
110 | instantiation | 119, 120, 121 | ⊢ |
| : , : , : |
111 | instantiation | 149, 144, 122 | ⊢ |
| : , : , : |
112 | instantiation | 149, 144, 123 | ⊢ |
| : , : , : |
113 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.nonzero_rational_is_rational_nonzero |
114 | assumption | | ⊢ |
115 | instantiation | 124, 136, 125 | ⊢ |
| : , : |
116 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero |
117 | instantiation | 149, 142, 126 | ⊢ |
| : , : , : |
118 | instantiation | 127, 138 | ⊢ |
| : |
119 | axiom | | ⊢ |
| proveit.logic.equality.equals_transitivity |
120 | instantiation | 128, 129 | ⊢ |
| : , : , : |
121 | instantiation | 130, 138 | ⊢ |
| : |
122 | instantiation | 152, 153, 131 | ⊢ |
| : , : , : |
123 | instantiation | 149, 132, 133 | ⊢ |
| : , : , : |
124 | theorem | | ⊢ |
| proveit.numbers.exponentiation.exp_rational_non_zero__not_zero |
125 | instantiation | 134, 135, 136 | ⊢ |
| : , : |
126 | instantiation | 149, 150, 154 | ⊢ |
| : , : , : |
127 | theorem | | ⊢ |
| proveit.numbers.multiplication.elim_one_left |
128 | axiom | | ⊢ |
| proveit.logic.equality.substitution |
129 | instantiation | 137, 138 | ⊢ |
| : |
130 | theorem | | ⊢ |
| proveit.numbers.division.frac_one_denom |
131 | assumption | | ⊢ |
132 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.rational_within_real |
133 | instantiation | 149, 139, 140 | ⊢ |
| : , : , : |
134 | theorem | | ⊢ |
| proveit.numbers.division.div_rational_nonzero_closure |
135 | instantiation | 149, 142, 141 | ⊢ |
| : , : , : |
136 | instantiation | 149, 142, 143 | ⊢ |
| : , : , : |
137 | theorem | | ⊢ |
| proveit.numbers.multiplication.elim_one_right |
138 | instantiation | 149, 144, 145 | ⊢ |
| : , : , : |
139 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.int_within_rational |
140 | instantiation | 149, 146, 147 | ⊢ |
| : , : , : |
141 | instantiation | 149, 150, 148 | ⊢ |
| : , : , : |
142 | theorem | | ⊢ |
| proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero |
143 | instantiation | 149, 150, 151 | ⊢ |
| : , : , : |
144 | theorem | | ⊢ |
| proveit.numbers.number_sets.complex_numbers.real_within_complex |
145 | instantiation | 152, 153, 154 | ⊢ |
| : , : , : |
146 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.nat_within_int |
147 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.nat1 |
148 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.posnat1 |
149 | theorem | | ⊢ |
| proveit.logic.sets.inclusion.superset_membership_from_proper_subset |
150 | theorem | | ⊢ |
| proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int |
151 | theorem | | ⊢ |
| proveit.numbers.numerals.decimals.posnat2 |
152 | theorem | | ⊢ |
| proveit.logic.sets.inclusion.unfold_subset_eq |
153 | instantiation | 155, 156 | ⊢ |
| : , : |
154 | assumption | | ⊢ |
155 | theorem | | ⊢ |
| proveit.logic.sets.inclusion.relax_proper_subset |
156 | theorem | | ⊢ |
| proveit.numbers.number_sets.real_numbers.nat_pos_within_real |
*equality replacement requirements |