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