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Show the Proof

In [1]:
import proveit
# Automation is not needed when only showing a stored proof:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%show_proof
Out[1]:
 step typerequirementsstatement
0instantiation1, 2  ⊢  
  : , :
1theorem  ⊢  
 proveit.logic.sets.membership.fold_not_in_set
2instantiation3, 4, 5, 6  ⊢  
  : , :
3theorem  ⊢  
 proveit.logic.booleans.implication.modus_tollens_denial
4instantiation7  ⊢  
  :
5deduction8  ⊢  
6theorem  ⊢  
 proveit.logic.booleans.negation.not_false
7theorem  ⊢  
 proveit.numbers.number_sets.rational_numbers.rational_membership_is_bool
8modus ponens9, 10  ⊢  
9instantiation11, 12, 13, 151  ⊢  
  : , : , : , : , : , :
10instantiation23, 14, 15  ⊢  
  : , :
11theorem  ⊢  
 proveit.logic.booleans.quantification.existence.skolem_elim
12instantiation16  ⊢  
  : , :
13instantiation16  ⊢  
  : , :
14instantiation17, 94  ⊢  
  :
15generalization18  ⊢  
16theorem  ⊢  
 proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
17theorem  ⊢  
 proveit.numbers.number_sets.rational_numbers.reduced_nat_pos_ratio
18deduction19, ,  ⊢  
19instantiation20, 21, 22, , ,  ⊢  
  :
20theorem  ⊢  
 proveit.logic.booleans.negation.negation_contradiction
21instantiation23, 42, 24, , ,  ⊢  
  : , :
22instantiation25, 151, 26, ,  ⊢  
  :
23theorem  ⊢  
 proveit.logic.booleans.conjunction.and_if_both
24instantiation43, 27, 75, 28, , ,  ⊢  
  : , :
25instantiation29, 131, 154, 30, ,  ⊢  
  : , :
26theorem  ⊢  
 proveit.numbers.numerals.decimals.less_1_2
27instantiation149, 54, 154  ⊢  
  : , : , :
28instantiation31, 49, 32, 33, 51, , ,  ⊢  
  : , : , :
29theorem  ⊢  
 proveit.numbers.divisibility.GCD_one_def
30instantiation34, 79  ⊢  
  : , :
31theorem  ⊢  
 proveit.numbers.divisibility.common_factor_elimination
32instantiation149, 144, 35  ⊢  
  : , : , :
33instantiation86, 36, 37, , ,  ⊢  
  : , : , :
34theorem  ⊢  
 proveit.logic.booleans.conjunction.right_from_and
35instantiation152, 153, 55  ⊢  
  : , : , :
36instantiation38, 39, 47, , ,  ⊢  
  : , : , :
37instantiation40, 49  ⊢  
  :
38theorem  ⊢  
 proveit.logic.equality.sub_left_side_into
39instantiation41, 75, 44, 151, 42, , ,  ⊢  
  : , : , :
40theorem  ⊢  
 proveit.numbers.exponentiation.square_expansion
41theorem  ⊢  
 proveit.numbers.divisibility.common_exponent_introduction
42instantiation43, 44, 75, 45, , ,  ⊢  
  : , :
43theorem  ⊢  
 proveit.numbers.divisibility.even__if__power_is_even
44instantiation149, 54, 131  ⊢  
  : , : , :
45instantiation86, 46, 47, , ,  ⊢  
  : , : , :
46instantiation48, 49, 50, 51  ⊢  
  : , :
47instantiation86, 52, 53, , ,  ⊢  
  : , : , :
48theorem  ⊢  
 proveit.numbers.divisibility.left_factor_divisibility
49instantiation149, 144, 57  ⊢  
  : , : , :
50instantiation149, 54, 55  ⊢  
  : , : , :
51instantiation85, 151  ⊢  
  :
52instantiation56, 57, 58, 122, 59, , ,  ⊢  
  : , : , :
53instantiation60, 61, 138, 151, 62*,  ⊢  
  : , : , :
54theorem  ⊢  
 proveit.numbers.number_sets.integers.nat_pos_within_int
55instantiation63, 64, 96  ⊢  
  : , :
56theorem  ⊢  
 proveit.numbers.exponentiation.exp_eq_real
57instantiation149, 132, 65  ⊢  
  : , : , :
58instantiation66, 71, 145,  ⊢  
  : , :
59instantiation67, 145, 71, 68, 69, 70*, , ,  ⊢  
  : , : , :
60theorem  ⊢  
 proveit.numbers.exponentiation.posnat_power_of_product
61instantiation149, 144, 71  ⊢  
  : , : , :
62instantiation72, 114, 73*  ⊢  
  :
63theorem  ⊢  
 proveit.numbers.exponentiation.exp_natpos_closure
64instantiation149, 74, 154  ⊢  
  : , : , :
65instantiation149, 139, 75  ⊢  
  : , : , :
66theorem  ⊢  
 proveit.numbers.multiplication.mult_real_closure_bin
67theorem  ⊢  
 proveit.numbers.multiplication.right_mult_eq_real
68instantiation76, 122, 145, 77,  ⊢  
  : , :
69instantiation78, 79  ⊢  
  : , :
70instantiation86, 80, 81,  ⊢  
  : , : , :
71instantiation149, 132, 82  ⊢  
  : , : , :
72theorem  ⊢  
 proveit.numbers.exponentiation.square_abs_rational_simp
73instantiation83, 151, 84  ⊢  
  : , :
74theorem  ⊢  
 proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
75instantiation149, 146, 96  ⊢  
  : , : , :
76theorem  ⊢  
 proveit.numbers.division.div_real_closure
77instantiation85, 154  ⊢  
  :
78theorem  ⊢  
 proveit.logic.booleans.conjunction.left_from_and
79assumption  ⊢  
80instantiation86, 87, 88,  ⊢  
  : , : , :
81instantiation89, 90, 91, 92  ⊢  
  : , : , : , :
82instantiation149, 93, 94  ⊢  
  : , : , :
83theorem  ⊢  
 proveit.numbers.exponentiation.nth_power_of_nth_root
84instantiation95, 96  ⊢  
  :
85theorem  ⊢  
 proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
86theorem  ⊢  
 proveit.logic.equality.sub_right_side_into
87instantiation97, 111, 112, 98, 99,  ⊢  
  : , : , : , : , :
88instantiation119, 100, 101  ⊢  
  : , : , :
89theorem  ⊢  
 proveit.logic.equality.four_chain_transitivity
90instantiation128, 102  ⊢  
  : , : , :
91instantiation128, 103  ⊢  
  : , : , :
92instantiation137, 111  ⊢  
  :
93theorem  ⊢  
 proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
94instantiation104, 105  ⊢  
  :
95theorem  ⊢  
 proveit.numbers.number_sets.natural_numbers.natural_lower_bound
96theorem  ⊢  
 proveit.numbers.numerals.decimals.nat2
97theorem  ⊢  
 proveit.numbers.division.mult_frac_cancel_denom_left
98instantiation149, 107, 106  ⊢  
  : , : , :
99instantiation149, 107, 108  ⊢  
  : , : , :
100instantiation128, 109  ⊢  
  : , : , :
101instantiation128, 110  ⊢  
  : , : , :
102instantiation130, 111  ⊢  
  :
103instantiation130, 112  ⊢  
  :
104theorem  ⊢  
 proveit.numbers.absolute_value.abs_rational_nonzero_closure
105instantiation113, 114, 115  ⊢  
  :
106instantiation149, 116, 135  ⊢  
  : , : , :
107theorem  ⊢  
 proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
108instantiation149, 116, 117  ⊢  
  : , : , :
109instantiation128, 118  ⊢  
  : , : , :
110instantiation119, 120, 121  ⊢  
  : , : , :
111instantiation149, 144, 122  ⊢  
  : , : , :
112instantiation149, 144, 123  ⊢  
  : , : , :
113theorem  ⊢  
 proveit.numbers.number_sets.rational_numbers.nonzero_rational_is_rational_nonzero
114assumption  ⊢  
115instantiation124, 136, 125  ⊢  
  : , :
116theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
117instantiation149, 142, 126  ⊢  
  : , : , :
118instantiation127, 138  ⊢  
  :
119axiom  ⊢  
 proveit.logic.equality.equals_transitivity
120instantiation128, 129  ⊢  
  : , : , :
121instantiation130, 138  ⊢  
  :
122instantiation152, 153, 131  ⊢  
  : , : , :
123instantiation149, 132, 133  ⊢  
  : , : , :
124theorem  ⊢  
 proveit.numbers.exponentiation.exp_rational_non_zero__not_zero
125instantiation134, 135, 136  ⊢  
  : , :
126instantiation149, 150, 154  ⊢  
  : , : , :
127theorem  ⊢  
 proveit.numbers.multiplication.elim_one_left
128axiom  ⊢  
 proveit.logic.equality.substitution
129instantiation137, 138  ⊢  
  :
130theorem  ⊢  
 proveit.numbers.division.frac_one_denom
131assumption  ⊢  
132theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.rational_within_real
133instantiation149, 139, 140  ⊢  
  : , : , :
134theorem  ⊢  
 proveit.numbers.division.div_rational_nonzero_closure
135instantiation149, 142, 141  ⊢  
  : , : , :
136instantiation149, 142, 143  ⊢  
  : , : , :
137theorem  ⊢  
 proveit.numbers.multiplication.elim_one_right
138instantiation149, 144, 145  ⊢  
  : , : , :
139theorem  ⊢  
 proveit.numbers.number_sets.rational_numbers.int_within_rational
140instantiation149, 146, 147  ⊢  
  : , : , :
141instantiation149, 150, 148  ⊢  
  : , : , :
142theorem  ⊢  
 proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
143instantiation149, 150, 151  ⊢  
  : , : , :
144theorem  ⊢  
 proveit.numbers.number_sets.complex_numbers.real_within_complex
145instantiation152, 153, 154  ⊢  
  : , : , :
146theorem  ⊢  
 proveit.numbers.number_sets.integers.nat_within_int
147theorem  ⊢  
 proveit.numbers.numerals.decimals.nat1
148theorem  ⊢  
 proveit.numbers.numerals.decimals.posnat1
149theorem  ⊢  
 proveit.logic.sets.inclusion.superset_membership_from_proper_subset
150theorem  ⊢  
 proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
151theorem  ⊢  
 proveit.numbers.numerals.decimals.posnat2
152theorem  ⊢  
 proveit.logic.sets.inclusion.unfold_subset_eq
153instantiation155, 156  ⊢  
  : , :
154assumption  ⊢  
155theorem  ⊢  
 proveit.logic.sets.inclusion.relax_proper_subset
156theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.nat_pos_within_real
*equality replacement requirements