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