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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
0instantiation1, 2, 3, 4, 5, 6*  ⊢  
  : , : , :
1theorem  ⊢  
 proveit.physics.quantum.algebra.scalar_mult_factorization
2theorem  ⊢  
 proveit.numbers.numerals.decimals.posnat3
3reference9  ⊢  
4instantiation124  ⊢  
  : , : , :
5instantiation19, 7, 16  ⊢  
  : , : , :
6instantiation8, 9, 10  ⊢  
  : , :
7instantiation11, 51, 52, 23, 24  ⊢  
  : , : , : , :
8axiom  ⊢  
 proveit.linear_algebra.scalar_multiplication.scalar_mult_extends_number_mult
9instantiation28, 12, 13, 14  ⊢  
  : , :
10instantiation19, 15, 16  ⊢  
  : , : , :
11theorem  ⊢  
 proveit.physics.quantum.algebra.qmult_op_ket_in_QmultCodomain
12instantiation161, 136, 17  ⊢  
  : , : , :
13instantiation71, 122, 18  ⊢  
  : , :
14instantiation19, 20, 21  ⊢  
  : , : , :
15instantiation22, 51, 52, 23, 24  ⊢  
  : , : , : , :
16instantiation25, 128, 26  ⊢  
  : , : , :
17instantiation161, 139, 27  ⊢  
  : , : , :
18instantiation28, 63, 122, 38  ⊢  
  : , :
19theorem  ⊢  
 proveit.logic.equality.sub_left_side_into
20instantiation29, 99, 30  ⊢  
  : , :
21instantiation60, 31  ⊢  
  : , : , :
22theorem  ⊢  
 proveit.physics.quantum.algebra.qmult_op_ket_is_ket
23instantiation50, 51, 52, 32  ⊢  
  : , : , :
24modus ponens33, 34  ⊢  
25axiom  ⊢  
 proveit.physics.quantum.algebra.multi_qmult_def
26instantiation132  ⊢  
  : , :
27instantiation161, 144, 157  ⊢  
  : , : , :
28theorem  ⊢  
 proveit.numbers.division.div_complex_closure
29theorem  ⊢  
 proveit.numbers.exponentiation.exp_rational_non_zero__not_zero
30instantiation161, 35, 36  ⊢  
  : , : , :
31instantiation37, 63, 122, 38, 39*  ⊢  
  : , :
32instantiation40, 51, 52, 41, 42  ⊢  
  : , : , : , : , :
33instantiation43, 126, 57  ⊢  
  : , : , : , : , : , :
34generalization44  ⊢  
35theorem  ⊢  
 proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational_nonzero
36instantiation45, 103, 46  ⊢  
  : , :
37theorem  ⊢  
 proveit.numbers.division.div_as_mult
38instantiation47, 128  ⊢  
  :
39instantiation95, 48, 49  ⊢  
  : , : , :
40theorem  ⊢  
 proveit.physics.quantum.algebra.qmult_op_op_is_op
41instantiation50, 51, 52, 53  ⊢  
  : , : , :
42instantiation54, 81, 55  ⊢  
  : , : , :
43theorem  ⊢  
 proveit.linear_algebra.addition.summation_closure
44instantiation56, 57, 58, 59  ⊢  
  : , : , : , :
45theorem  ⊢  
 proveit.numbers.multiplication.mult_rational_pos_closure_bin
46instantiation161, 127, 160  ⊢  
  : , : , :
47theorem  ⊢  
 proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
48instantiation60, 61  ⊢  
  : , : , :
49instantiation62, 63, 64  ⊢  
  : , :
50theorem  ⊢  
 proveit.physics.quantum.algebra.qmult_op_is_linmap
51instantiation65, 81  ⊢  
  :
52theorem  ⊢  
 proveit.linear_algebra.inner_products.complex_set_is_hilbert_space
53instantiation66, 160, 67  ⊢  
  : , :
54theorem  ⊢  
 proveit.physics.quantum.algebra.qmult_matrix_is_linmap
55instantiation89, 68, 69  ⊢  
  : , : , :
56theorem  ⊢  
 proveit.linear_algebra.scalar_multiplication.scalar_mult_closure
57instantiation70, 81  ⊢  
  :
58instantiation71, 72, 73  ⊢  
  : , :
59instantiation74, 160, 146  ⊢  
  : , :
60axiom  ⊢  
 proveit.logic.equality.substitution
61instantiation75, 76, 126, 77*  ⊢  
  : , :
62theorem  ⊢  
 proveit.numbers.multiplication.commutation
63instantiation161, 136, 78  ⊢  
  : , : , :
64instantiation161, 136, 79  ⊢  
  : , : , :
65theorem  ⊢  
 proveit.linear_algebra.inner_products.complex_vec_set_is_hilbert_space
66theorem  ⊢  
 proveit.physics.quantum.algebra.num_bra_is_lin_map
67assumption  ⊢  
68instantiation80, 81  ⊢  
  :
69instantiation82, 160  ⊢  
  :
70theorem  ⊢  
 proveit.linear_algebra.complex_vec_set_is_vec_space
71theorem  ⊢  
 proveit.numbers.exponentiation.exp_complex_closure
72instantiation161, 136, 83  ⊢  
  : , : , :
73instantiation104, 84, 85  ⊢  
  : , : , :
74theorem  ⊢  
 proveit.physics.quantum.algebra.num_ket_in_register_space
75theorem  ⊢  
 proveit.numbers.exponentiation.neg_power_as_div
76instantiation161, 86, 87  ⊢  
  : , : , :
77instantiation88, 122  ⊢  
  :
78instantiation89, 90, 160  ⊢  
  : , : , :
79instantiation161, 139, 91  ⊢  
  : , : , :
80theorem  ⊢  
 proveit.linear_algebra.matrices.unitaries_are_matrices
81instantiation92, 158, 155  ⊢  
  : , :
82theorem  ⊢  
 proveit.physics.quantum.QFT.invFT_is_unitary
83instantiation161, 141, 93  ⊢  
  : , : , :
84instantiation129, 107, 94  ⊢  
  : , :
85instantiation95, 96, 97  ⊢  
  : , : , :
86theorem  ⊢  
 proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
87instantiation161, 98, 99  ⊢  
  : , : , :
88theorem  ⊢  
 proveit.numbers.exponentiation.complex_x_to_first_power_is_x
89theorem  ⊢  
 proveit.logic.sets.inclusion.unfold_subset_eq
90instantiation100, 101  ⊢  
  : , :
91instantiation161, 102, 103  ⊢  
  : , : , :
92theorem  ⊢  
 proveit.numbers.exponentiation.exp_natpos_closure
93theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.e_is_real_pos
94instantiation104, 105, 106  ⊢  
  : , : , :
95axiom  ⊢  
 proveit.logic.equality.equals_transitivity
96instantiation117, 163, 108, 118, 110, 119, 107, 130, 131, 121  ⊢  
  : , : , : , : , : , :
97instantiation117, 118, 158, 108, 119, 109, 110, 122, 123, 130, 131, 121  ⊢  
  : , : , : , : , : , :
98theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
99instantiation161, 111, 112  ⊢  
  : , : , :
100theorem  ⊢  
 proveit.logic.sets.inclusion.relax_proper_subset
101theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.nat_pos_within_real
102theorem  ⊢  
 proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
103instantiation113, 114, 115  ⊢  
  : , :
104theorem  ⊢  
 proveit.logic.equality.sub_right_side_into
105instantiation129, 116, 121  ⊢  
  : , :
106instantiation117, 118, 158, 163, 119, 120, 130, 131, 121  ⊢  
  : , : , : , : , : , :
107instantiation129, 122, 123  ⊢  
  : , :
108theorem  ⊢  
 proveit.numbers.numerals.decimals.nat3
109instantiation132  ⊢  
  : , :
110instantiation124  ⊢  
  : , : , :
111theorem  ⊢  
 proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
112instantiation161, 125, 128  ⊢  
  : , : , :
113theorem  ⊢  
 proveit.numbers.division.div_rational_pos_closure
114instantiation161, 127, 126  ⊢  
  : , : , :
115instantiation161, 127, 128  ⊢  
  : , : , :
116instantiation129, 130, 131  ⊢  
  : , :
117theorem  ⊢  
 proveit.numbers.multiplication.disassociation
118axiom  ⊢  
 proveit.numbers.number_sets.natural_numbers.zero_in_nats
119theorem  ⊢  
 proveit.core_expr_types.tuples.tuple_len_0_typical_eq
120instantiation132  ⊢  
  : , :
121instantiation161, 136, 133  ⊢  
  : , : , :
122instantiation161, 136, 134  ⊢  
  : , : , :
123instantiation161, 136, 135  ⊢  
  : , : , :
124theorem  ⊢  
 proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
125theorem  ⊢  
 proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
126theorem  ⊢  
 proveit.numbers.numerals.decimals.posnat1
127theorem  ⊢  
 proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
128theorem  ⊢  
 proveit.numbers.numerals.decimals.posnat2
129theorem  ⊢  
 proveit.numbers.multiplication.mult_complex_closure_bin
130theorem  ⊢  
 proveit.numbers.number_sets.complex_numbers.i_is_complex
131instantiation161, 136, 137  ⊢  
  : , : , :
132theorem  ⊢  
 proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
133instantiation161, 139, 138  ⊢  
  : , : , :
134instantiation161, 139, 140  ⊢  
  : , : , :
135instantiation161, 141, 142  ⊢  
  : , : , :
136theorem  ⊢  
 proveit.numbers.number_sets.complex_numbers.real_within_complex
137theorem  ⊢  
 proveit.physics.quantum.QPE._phase_is_real
138instantiation161, 144, 143  ⊢  
  : , : , :
139theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.rational_within_real
140instantiation161, 144, 154  ⊢  
  : , : , :
141theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.real_pos_within_real
142theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.pi_is_real_pos
143instantiation161, 145, 146  ⊢  
  : , : , :
144theorem  ⊢  
 proveit.numbers.number_sets.rational_numbers.int_within_rational
145instantiation147, 148, 149  ⊢  
  : , :
146assumption  ⊢  
147theorem  ⊢  
 proveit.numbers.number_sets.integers.int_interval_within_int
148theorem  ⊢  
 proveit.numbers.number_sets.integers.zero_is_int
149instantiation150, 151, 152  ⊢  
  : , :
150theorem  ⊢  
 proveit.numbers.addition.add_int_closure_bin
151instantiation153, 154, 155  ⊢  
  : , :
152instantiation156, 157  ⊢  
  :
153theorem  ⊢  
 proveit.numbers.exponentiation.exp_int_closure
154instantiation161, 162, 158  ⊢  
  : , : , :
155instantiation161, 159, 160  ⊢  
  : , : , :
156theorem  ⊢  
 proveit.numbers.negation.int_closure
157instantiation161, 162, 163  ⊢  
  : , : , :
158theorem  ⊢  
 proveit.numbers.numerals.decimals.nat2
159theorem  ⊢  
 proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
160axiom  ⊢  
 proveit.physics.quantum.QPE._t_in_natural_pos
161theorem  ⊢  
 proveit.logic.sets.inclusion.superset_membership_from_proper_subset
162theorem  ⊢  
 proveit.numbers.number_sets.integers.nat_within_int
163theorem  ⊢  
 proveit.numbers.numerals.decimals.nat1
*equality replacement requirements