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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, 3, 4, 5, 6, 7, 8, 9, 10*  ⊢  
  : , : , : , : , : , :
1theorem  ⊢  
 proveit.numbers.multiplication.association
2axiom  ⊢  
 proveit.numbers.number_sets.natural_numbers.zero_in_nats
3reference123  ⊢  
4reference113  ⊢  
5theorem  ⊢  
 proveit.core_expr_types.tuples.tuple_len_0_typical_eq
6instantiation51  ⊢  
  : , :
7instantiation121, 69, 11  ⊢  
  : , : , :
8reference58  ⊢  
9instantiation121, 69, 12  ⊢  
  : , : , :
10instantiation13, 40, 58, 14, 15, 16*, 17*  ⊢  
  : , : , : , :
11instantiation121, 76, 18  ⊢  
  : , : , :
12modus ponens19, 20  ⊢  
13theorem  ⊢  
 proveit.numbers.division.prod_of_fracs
14instantiation121, 54, 21  ⊢  
  : , : , :
15instantiation121, 54, 22  ⊢  
  : , : , :
16instantiation23, 58  ⊢  
  :
17instantiation24, 25, 26  ⊢  
  : , : , :
18instantiation121, 27, 28  ⊢  
  : , : , :
19instantiation29  ⊢  
  : , : , :
20generalization30  ⊢  
21instantiation121, 66, 31  ⊢  
  : , : , :
22instantiation121, 66, 32  ⊢  
  : , : , :
23theorem  ⊢  
 proveit.numbers.division.frac_one_denom
24axiom  ⊢  
 proveit.logic.equality.equals_transitivity
25instantiation33, 123, 34, 35, 36, 37  ⊢  
  : , : , : , :
26instantiation38, 39, 40, 41*, 42*  ⊢  
  : , : , :
27theorem  ⊢  
 proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
28instantiation43, 44, 45  ⊢  
  : , :
29theorem  ⊢  
 proveit.numbers.summation.summation_real_closure
30instantiation46, 56, 47, 48,  ⊢  
  : , :
31instantiation121, 74, 49  ⊢  
  : , : , :
32instantiation121, 74, 50  ⊢  
  : , : , :
33axiom  ⊢  
 proveit.core_expr_types.operations.operands_substitution
34instantiation51  ⊢  
  : , :
35instantiation51  ⊢  
  : , :
36instantiation52, 58  ⊢  
  :
37instantiation57, 53  ⊢  
  :
38theorem  ⊢  
 proveit.numbers.division.frac_cancel_left
39instantiation121, 54, 55  ⊢  
  : , : , :
40instantiation121, 69, 56  ⊢  
  : , : , :
41instantiation57, 58  ⊢  
  :
42theorem  ⊢  
 proveit.numbers.numerals.decimals.mult_2_2
43theorem  ⊢  
 proveit.numbers.division.div_rational_pos_closure
44instantiation121, 59, 107  ⊢  
  : , : , :
45instantiation121, 59, 64  ⊢  
  : , : , :
46theorem  ⊢  
 proveit.numbers.division.div_real_closure
47instantiation60, 61, 123,  ⊢  
  : , :
48instantiation62, 63, 67,  ⊢  
  : , :
49instantiation121, 80, 64  ⊢  
  : , : , :
50instantiation121, 80, 107  ⊢  
  : , : , :
51theorem  ⊢  
 proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
52theorem  ⊢  
 proveit.numbers.multiplication.elim_one_left
53instantiation121, 69, 65  ⊢  
  : , : , :
54theorem  ⊢  
 proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
55instantiation121, 66, 67  ⊢  
  : , : , :
56instantiation121, 76, 68  ⊢  
  : , : , :
57theorem  ⊢  
 proveit.numbers.multiplication.elim_one_right
58instantiation121, 69, 70  ⊢  
  : , : , :
59theorem  ⊢  
 proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
60theorem  ⊢  
 proveit.numbers.exponentiation.exp_real_closure_nat_power
61instantiation121, 76, 71,  ⊢  
  : , : , :
62theorem  ⊢  
 proveit.numbers.exponentiation.exp_rational_non_zero__not_zero
63instantiation121, 74, 72,  ⊢  
  : , : , :
64theorem  ⊢  
 proveit.numbers.numerals.decimals.posnat4
65instantiation121, 76, 73  ⊢  
  : , : , :
66theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
67instantiation121, 74, 75  ⊢  
  : , : , :
68instantiation121, 82, 110  ⊢  
  : , : , :
69theorem  ⊢  
 proveit.numbers.number_sets.complex_numbers.real_within_complex
70instantiation121, 76, 77  ⊢  
  : , : , :
71instantiation121, 82, 83,  ⊢  
  : , : , :
72instantiation121, 80, 78,  ⊢  
  : , : , :
73instantiation121, 82, 79  ⊢  
  : , : , :
74theorem  ⊢  
 proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
75instantiation121, 80, 81  ⊢  
  : , : , :
76theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.rational_within_real
77instantiation121, 82, 120  ⊢  
  : , : , :
78instantiation97, 83, 84,  ⊢  
  :
79instantiation121, 122, 85  ⊢  
  : , : , :
80theorem  ⊢  
 proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
81theorem  ⊢  
 proveit.numbers.numerals.decimals.posnat2
82theorem  ⊢  
 proveit.numbers.number_sets.rational_numbers.int_within_rational
83instantiation121, 86, 93,  ⊢  
  : , : , :
84instantiation103, 87, 88,  ⊢  
  : , : , :
85theorem  ⊢  
 proveit.numbers.numerals.decimals.nat4
86instantiation108, 91, 92  ⊢  
  : , :
87instantiation89, 90  ⊢  
  :
88instantiation109, 91, 92, 93,  ⊢  
  : , : , :
89theorem  ⊢  
 proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
90instantiation94, 95, 107  ⊢  
  : , :
91instantiation114, 98, 110  ⊢  
  : , :
92instantiation114, 115, 96  ⊢  
  : , :
93assumption  ⊢  
94theorem  ⊢  
 proveit.numbers.addition.add_nat_pos_closure_bin
95instantiation97, 98, 99  ⊢  
  :
96instantiation121, 100, 101  ⊢  
  : , : , :
97theorem  ⊢  
 proveit.numbers.number_sets.integers.pos_int_is_natural_pos
98instantiation121, 102, 112  ⊢  
  : , : , :
99instantiation103, 104, 105  ⊢  
  : , : , :
100theorem  ⊢  
 proveit.numbers.number_sets.integers.neg_int_within_int
101instantiation106, 107  ⊢  
  :
102instantiation108, 110, 111  ⊢  
  : , :
103theorem  ⊢  
 proveit.numbers.ordering.transitivity_less_less_eq
104theorem  ⊢  
 proveit.numbers.numerals.decimals.less_0_1
105instantiation109, 110, 111, 112  ⊢  
  : , : , :
106theorem  ⊢  
 proveit.numbers.negation.int_neg_closure
107theorem  ⊢  
 proveit.numbers.numerals.decimals.posnat1
108theorem  ⊢  
 proveit.numbers.number_sets.integers.int_interval_within_int
109theorem  ⊢  
 proveit.numbers.number_sets.integers.interval_lower_bound
110instantiation121, 122, 113  ⊢  
  : , : , :
111instantiation114, 115, 116  ⊢  
  : , :
112assumption  ⊢  
113theorem  ⊢  
 proveit.numbers.numerals.decimals.nat1
114theorem  ⊢  
 proveit.numbers.addition.add_int_closure_bin
115instantiation121, 117, 118  ⊢  
  : , : , :
116instantiation119, 120  ⊢  
  :
117theorem  ⊢  
 proveit.numbers.number_sets.integers.nat_pos_within_int
118theorem  ⊢  
 proveit.physics.quantum.QPE._two_pow_t_minus_one_is_nat_pos
119theorem  ⊢  
 proveit.numbers.negation.int_closure
120instantiation121, 122, 123  ⊢  
  : , : , :
121theorem  ⊢  
 proveit.logic.sets.inclusion.superset_membership_from_proper_subset
122theorem  ⊢  
 proveit.numbers.number_sets.integers.nat_within_int
123theorem  ⊢  
 proveit.numbers.numerals.decimals.nat2
*equality replacement requirements