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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
0generalization1  ⊢  
1instantiation45, 2,  ⊢  
  : , :
2instantiation3, 4, 5, 6, 7,  ⊢  
  : , : , :
3theorem  ⊢  
 proveit.numbers.division.strong_div_from_denom_bound__all_pos
4instantiation136, 9, 8  ⊢  
  : , : , :
5instantiation136, 9, 10,  ⊢  
  : , : , :
6instantiation11, 28, 12,  ⊢  
  :
7instantiation13, 102, 14, 28, 15, 16,  ⊢  
  : , : , :
8instantiation136, 17, 72  ⊢  
  : , : , :
9theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
10instantiation136, 17, 18,  ⊢  
  : , : , :
11theorem  ⊢  
 proveit.numbers.exponentiation.sqrd_pos_closure
12instantiation19, 20,  ⊢  
  : , :
13theorem  ⊢  
 proveit.numbers.exponentiation.exp_pos_less
14instantiation35, 36, 40,  ⊢  
  : , :
15instantiation21, 34, 22,  ⊢  
  : , :
16instantiation23, 24  ⊢  
  :
17theorem  ⊢  
 proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
18instantiation25, 26, 138,  ⊢  
  : , :
19theorem  ⊢  
 proveit.logic.equality.not_equals_symmetry
20instantiation27, 84, 28, 29,  ⊢  
  : , :
21theorem  ⊢  
 proveit.logic.booleans.conjunction.and_if_both
22instantiation30, 36, 40, 37, 31,  ⊢  
  : , : , :
23theorem  ⊢  
 proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
24theorem  ⊢  
 proveit.numbers.numerals.decimals.posnat2
25theorem  ⊢  
 proveit.numbers.exponentiation.exp_natpos_closure
26instantiation32, 33, 34,  ⊢  
  :
27theorem  ⊢  
 proveit.numbers.ordering.less_is_not_eq
28instantiation35, 36, 37,  ⊢  
  : , :
29instantiation60, 38,  ⊢  
  : , :
30theorem  ⊢  
 proveit.numbers.addition.strong_bound_via_right_term_bound
31instantiation39, 40, 48, 119, 50, 41, 42*, 43*  ⊢  
  : , : , :
32theorem  ⊢  
 proveit.numbers.number_sets.integers.nonneg_int_is_natural
33instantiation126, 62, 44,  ⊢  
  : , :
34instantiation45, 46,  ⊢  
  : , :
35theorem  ⊢  
 proveit.numbers.addition.add_real_closure_bin
36instantiation136, 124, 47,  ⊢  
  : , : , :
37instantiation51, 48  ⊢  
  :
38instantiation49, 50, 61,  ⊢  
  : , : , :
39theorem  ⊢  
 proveit.numbers.multiplication.reversed_strong_bound_via_right_factor_bound
40instantiation51, 119  ⊢  
  :
41instantiation52, 59  ⊢  
  :
42instantiation53, 110, 54*  ⊢  
  : , :
43instantiation55, 56, 57  ⊢  
  : , : , :
44instantiation136, 58, 59  ⊢  
  : , : , :
45theorem  ⊢  
 proveit.numbers.ordering.relax_less
46instantiation60, 61,  ⊢  
  : , :
47instantiation136, 129, 62,  ⊢  
  : , : , :
48instantiation63, 84, 119, 65  ⊢  
  : , : , :
49axiom  ⊢  
 proveit.numbers.ordering.transitivity_less_less
50instantiation64, 84, 119, 65  ⊢  
  : , : , :
51theorem  ⊢  
 proveit.numbers.negation.real_closure
52theorem  ⊢  
 proveit.numbers.number_sets.integers.negative_if_in_neg_int
53theorem  ⊢  
 proveit.numbers.multiplication.mult_neg_left
54instantiation66, 110  ⊢  
  :
55axiom  ⊢  
 proveit.logic.equality.equals_transitivity
56instantiation67, 135, 138, 78, 80, 79, 68, 81, 82  ⊢  
  : , : , : , : , : , :
57instantiation69, 78, 138, 79, 80, 110, 81, 82, 70*  ⊢  
  : , : , : , : , :
58theorem  ⊢  
 proveit.numbers.number_sets.integers.neg_int_within_int
59instantiation71, 72  ⊢  
  :
60theorem  ⊢  
 proveit.numbers.addition.subtraction.pos_difference
61instantiation96, 73, 74,  ⊢  
  : , : , :
62instantiation136, 75, 88,  ⊢  
  : , : , :
63theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.all_in_interval_co__is__real
64theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.interval_co_upper_bound
65theorem  ⊢  
 proveit.physics.quantum.QPE._scaled_delta_b_floor_in_interval
66theorem  ⊢  
 proveit.numbers.multiplication.elim_one_right
67theorem  ⊢  
 proveit.numbers.multiplication.disassociation
68instantiation76, 110  ⊢  
  :
69theorem  ⊢  
 proveit.numbers.multiplication.mult_neg_any
70instantiation77, 78, 138, 79, 80, 81, 82  ⊢  
  : , : , : , :
71theorem  ⊢  
 proveit.numbers.negation.int_neg_closure
72theorem  ⊢  
 proveit.numbers.numerals.decimals.posnat1
73instantiation83, 119, 84, 85, 86, 87*  ⊢  
  : , : , :
74instantiation107, 89, 127, 88,  ⊢  
  : , : , :
75instantiation122, 89, 127  ⊢  
  : , :
76theorem  ⊢  
 proveit.numbers.negation.complex_closure
77theorem  ⊢  
 proveit.numbers.multiplication.elim_one_any
78axiom  ⊢  
 proveit.numbers.number_sets.natural_numbers.zero_in_nats
79theorem  ⊢  
 proveit.core_expr_types.tuples.tuple_len_0_typical_eq
80instantiation90  ⊢  
  : , :
81instantiation91, 92, 93  ⊢  
  : , :
82instantiation136, 118, 94  ⊢  
  : , : , :
83theorem  ⊢  
 proveit.numbers.addition.strong_bound_via_left_term_bound
84theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.zero_is_real
85instantiation136, 124, 95  ⊢  
  : , : , :
86instantiation96, 97, 98  ⊢  
  : , : , :
87instantiation99, 100, 101  ⊢  
  : , : , :
88assumption  ⊢  
89instantiation126, 106, 130  ⊢  
  : , :
90theorem  ⊢  
 proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
91theorem  ⊢  
 proveit.numbers.exponentiation.exp_complex_closure
92instantiation136, 118, 102  ⊢  
  : , : , :
93instantiation136, 118, 103  ⊢  
  : , : , :
94instantiation104, 105  ⊢  
  :
95instantiation136, 129, 106  ⊢  
  : , : , :
96theorem  ⊢  
 proveit.numbers.ordering.transitivity_less_less_eq
97theorem  ⊢  
 proveit.numbers.numerals.decimals.less_0_1
98instantiation107, 130, 123, 117  ⊢  
  : , : , :
99theorem  ⊢  
 proveit.logic.equality.sub_right_side_into
100instantiation108, 110  ⊢  
  :
101instantiation109, 110, 111  ⊢  
  : , :
102instantiation136, 124, 112  ⊢  
  : , : , :
103instantiation113, 114, 115  ⊢  
  : , : , :
104theorem  ⊢  
 proveit.physics.quantum.QPE._delta_b_is_real
105theorem  ⊢  
 proveit.physics.quantum.QPE._best_floor_is_int
106instantiation136, 116, 117  ⊢  
  : , : , :
107theorem  ⊢  
 proveit.numbers.number_sets.integers.interval_lower_bound
108theorem  ⊢  
 proveit.numbers.addition.elim_zero_right
109theorem  ⊢  
 proveit.numbers.addition.commutation
110instantiation136, 118, 119  ⊢  
  : , : , :
111theorem  ⊢  
 proveit.numbers.number_sets.complex_numbers.zero_is_complex
112instantiation136, 129, 134  ⊢  
  : , : , :
113theorem  ⊢  
 proveit.logic.sets.inclusion.unfold_subset_eq
114instantiation120, 121  ⊢  
  : , :
115axiom  ⊢  
 proveit.physics.quantum.QPE._t_in_natural_pos
116instantiation122, 130, 123  ⊢  
  : , :
117assumption  ⊢  
118theorem  ⊢  
 proveit.numbers.number_sets.complex_numbers.real_within_complex
119instantiation136, 124, 125  ⊢  
  : , : , :
120theorem  ⊢  
 proveit.logic.sets.inclusion.relax_proper_subset
121theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.nat_pos_within_real
122theorem  ⊢  
 proveit.numbers.number_sets.integers.int_interval_within_int
123instantiation126, 127, 128  ⊢  
  : , :
124theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.rational_within_real
125instantiation136, 129, 130  ⊢  
  : , : , :
126theorem  ⊢  
 proveit.numbers.addition.add_int_closure_bin
127instantiation136, 131, 132  ⊢  
  : , : , :
128instantiation133, 134  ⊢  
  :
129theorem  ⊢  
 proveit.numbers.number_sets.rational_numbers.int_within_rational
130instantiation136, 137, 135  ⊢  
  : , : , :
131theorem  ⊢  
 proveit.numbers.number_sets.integers.nat_pos_within_int
132theorem  ⊢  
 proveit.physics.quantum.QPE._two_pow_t_minus_one_is_nat_pos
133theorem  ⊢  
 proveit.numbers.negation.int_closure
134instantiation136, 137, 138  ⊢  
  : , : , :
135theorem  ⊢  
 proveit.numbers.numerals.decimals.nat1
136theorem  ⊢  
 proveit.logic.sets.inclusion.superset_membership_from_proper_subset
137theorem  ⊢  
 proveit.numbers.number_sets.integers.nat_within_int
138theorem  ⊢  
 proveit.numbers.numerals.decimals.nat2
*equality replacement requirements