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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  ⊢  
1instantiation2, 3, 4, 5, 6,  ⊢  
  : , : , :
2theorem  ⊢  
 proveit.numbers.division.strong_div_from_denom_bound__all_pos
3instantiation135, 8, 7  ⊢  
  : , : , :
4instantiation135, 8, 9,  ⊢  
  : , : , :
5instantiation10, 27, 11,  ⊢  
  :
6instantiation12, 101, 13, 27, 14, 15,  ⊢  
  : , : , :
7instantiation135, 16, 71  ⊢  
  : , : , :
8theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
9instantiation135, 16, 17,  ⊢  
  : , : , :
10theorem  ⊢  
 proveit.numbers.exponentiation.sqrd_pos_closure
11instantiation18, 19,  ⊢  
  : , :
12theorem  ⊢  
 proveit.numbers.exponentiation.exp_pos_less
13instantiation34, 35, 39,  ⊢  
  : , :
14instantiation20, 33, 21,  ⊢  
  : , :
15instantiation22, 23  ⊢  
  :
16theorem  ⊢  
 proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
17instantiation24, 25, 137,  ⊢  
  : , :
18theorem  ⊢  
 proveit.logic.equality.not_equals_symmetry
19instantiation26, 83, 27, 28,  ⊢  
  : , :
20theorem  ⊢  
 proveit.logic.booleans.conjunction.and_if_both
21instantiation29, 35, 39, 36, 30,  ⊢  
  : , : , :
22theorem  ⊢  
 proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
23theorem  ⊢  
 proveit.numbers.numerals.decimals.posnat2
24theorem  ⊢  
 proveit.numbers.exponentiation.exp_natpos_closure
25instantiation31, 32, 33,  ⊢  
  :
26theorem  ⊢  
 proveit.numbers.ordering.less_is_not_eq
27instantiation34, 35, 36,  ⊢  
  : , :
28instantiation59, 37,  ⊢  
  : , :
29theorem  ⊢  
 proveit.numbers.addition.strong_bound_via_right_term_bound
30instantiation38, 39, 47, 118, 49, 40, 41*, 42*  ⊢  
  : , : , :
31theorem  ⊢  
 proveit.numbers.number_sets.integers.nonneg_int_is_natural
32instantiation125, 61, 43,  ⊢  
  : , :
33instantiation44, 45,  ⊢  
  : , :
34theorem  ⊢  
 proveit.numbers.addition.add_real_closure_bin
35instantiation135, 123, 46,  ⊢  
  : , : , :
36instantiation50, 47  ⊢  
  :
37instantiation48, 49, 60,  ⊢  
  : , : , :
38theorem  ⊢  
 proveit.numbers.multiplication.reversed_strong_bound_via_right_factor_bound
39instantiation50, 118  ⊢  
  :
40instantiation51, 58  ⊢  
  :
41instantiation52, 109, 53*  ⊢  
  : , :
42instantiation54, 55, 56  ⊢  
  : , : , :
43instantiation135, 57, 58  ⊢  
  : , : , :
44theorem  ⊢  
 proveit.numbers.ordering.relax_less
45instantiation59, 60,  ⊢  
  : , :
46instantiation135, 128, 61,  ⊢  
  : , : , :
47instantiation62, 83, 118, 64  ⊢  
  : , : , :
48axiom  ⊢  
 proveit.numbers.ordering.transitivity_less_less
49instantiation63, 83, 118, 64  ⊢  
  : , : , :
50theorem  ⊢  
 proveit.numbers.negation.real_closure
51theorem  ⊢  
 proveit.numbers.number_sets.integers.negative_if_in_neg_int
52theorem  ⊢  
 proveit.numbers.multiplication.mult_neg_left
53instantiation65, 109  ⊢  
  :
54axiom  ⊢  
 proveit.logic.equality.equals_transitivity
55instantiation66, 134, 137, 77, 79, 78, 67, 80, 81  ⊢  
  : , : , : , : , : , :
56instantiation68, 77, 137, 78, 79, 109, 80, 81, 69*  ⊢  
  : , : , : , : , :
57theorem  ⊢  
 proveit.numbers.number_sets.integers.neg_int_within_int
58instantiation70, 71  ⊢  
  :
59theorem  ⊢  
 proveit.numbers.addition.subtraction.pos_difference
60instantiation95, 72, 73,  ⊢  
  : , : , :
61instantiation135, 74, 87,  ⊢  
  : , : , :
62theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.all_in_interval_co__is__real
63theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.interval_co_upper_bound
64theorem  ⊢  
 proveit.physics.quantum.QPE._scaled_delta_b_floor_in_interval
65theorem  ⊢  
 proveit.numbers.multiplication.elim_one_right
66theorem  ⊢  
 proveit.numbers.multiplication.disassociation
67instantiation75, 109  ⊢  
  :
68theorem  ⊢  
 proveit.numbers.multiplication.mult_neg_any
69instantiation76, 77, 137, 78, 79, 80, 81  ⊢  
  : , : , : , :
70theorem  ⊢  
 proveit.numbers.negation.int_neg_closure
71theorem  ⊢  
 proveit.numbers.numerals.decimals.posnat1
72instantiation82, 118, 83, 84, 85, 86*  ⊢  
  : , : , :
73instantiation106, 88, 126, 87,  ⊢  
  : , : , :
74instantiation121, 88, 126  ⊢  
  : , :
75theorem  ⊢  
 proveit.numbers.negation.complex_closure
76theorem  ⊢  
 proveit.numbers.multiplication.elim_one_any
77axiom  ⊢  
 proveit.numbers.number_sets.natural_numbers.zero_in_nats
78theorem  ⊢  
 proveit.core_expr_types.tuples.tuple_len_0_typical_eq
79instantiation89  ⊢  
  : , :
80instantiation90, 91, 92  ⊢  
  : , :
81instantiation135, 117, 93  ⊢  
  : , : , :
82theorem  ⊢  
 proveit.numbers.addition.strong_bound_via_left_term_bound
83theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.zero_is_real
84instantiation135, 123, 94  ⊢  
  : , : , :
85instantiation95, 96, 97  ⊢  
  : , : , :
86instantiation98, 99, 100  ⊢  
  : , : , :
87assumption  ⊢  
88instantiation125, 105, 129  ⊢  
  : , :
89theorem  ⊢  
 proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
90theorem  ⊢  
 proveit.numbers.exponentiation.exp_complex_closure
91instantiation135, 117, 101  ⊢  
  : , : , :
92instantiation135, 117, 102  ⊢  
  : , : , :
93instantiation103, 104  ⊢  
  :
94instantiation135, 128, 105  ⊢  
  : , : , :
95theorem  ⊢  
 proveit.numbers.ordering.transitivity_less_less_eq
96theorem  ⊢  
 proveit.numbers.numerals.decimals.less_0_1
97instantiation106, 129, 122, 116  ⊢  
  : , : , :
98theorem  ⊢  
 proveit.logic.equality.sub_right_side_into
99instantiation107, 109  ⊢  
  :
100instantiation108, 109, 110  ⊢  
  : , :
101instantiation135, 123, 111  ⊢  
  : , : , :
102instantiation112, 113, 114  ⊢  
  : , : , :
103theorem  ⊢  
 proveit.physics.quantum.QPE._delta_b_is_real
104theorem  ⊢  
 proveit.physics.quantum.QPE._best_floor_is_int
105instantiation135, 115, 116  ⊢  
  : , : , :
106theorem  ⊢  
 proveit.numbers.number_sets.integers.interval_lower_bound
107theorem  ⊢  
 proveit.numbers.addition.elim_zero_right
108theorem  ⊢  
 proveit.numbers.addition.commutation
109instantiation135, 117, 118  ⊢  
  : , : , :
110theorem  ⊢  
 proveit.numbers.number_sets.complex_numbers.zero_is_complex
111instantiation135, 128, 133  ⊢  
  : , : , :
112theorem  ⊢  
 proveit.logic.sets.inclusion.unfold_subset_eq
113instantiation119, 120  ⊢  
  : , :
114axiom  ⊢  
 proveit.physics.quantum.QPE._t_in_natural_pos
115instantiation121, 129, 122  ⊢  
  : , :
116assumption  ⊢  
117theorem  ⊢  
 proveit.numbers.number_sets.complex_numbers.real_within_complex
118instantiation135, 123, 124  ⊢  
  : , : , :
119theorem  ⊢  
 proveit.logic.sets.inclusion.relax_proper_subset
120theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.nat_pos_within_real
121theorem  ⊢  
 proveit.numbers.number_sets.integers.int_interval_within_int
122instantiation125, 126, 127  ⊢  
  : , :
123theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.rational_within_real
124instantiation135, 128, 129  ⊢  
  : , : , :
125theorem  ⊢  
 proveit.numbers.addition.add_int_closure_bin
126instantiation135, 130, 131  ⊢  
  : , : , :
127instantiation132, 133  ⊢  
  :
128theorem  ⊢  
 proveit.numbers.number_sets.rational_numbers.int_within_rational
129instantiation135, 136, 134  ⊢  
  : , : , :
130theorem  ⊢  
 proveit.numbers.number_sets.integers.nat_pos_within_int
131theorem  ⊢  
 proveit.physics.quantum.QPE._two_pow_t_minus_one_is_nat_pos
132theorem  ⊢  
 proveit.numbers.negation.int_closure
133instantiation135, 136, 137  ⊢  
  : , : , :
134theorem  ⊢  
 proveit.numbers.numerals.decimals.nat1
135theorem  ⊢  
 proveit.logic.sets.inclusion.superset_membership_from_proper_subset
136theorem  ⊢  
 proveit.numbers.number_sets.integers.nat_within_int
137theorem  ⊢  
 proveit.numbers.numerals.decimals.nat2
*equality replacement requirements