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