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 type | requirements | statement | ||
|---|---|---|---|---|
| 0 | instantiation | 1, 2, 3 |  ,  ,  ,  ,  ,  ,  ,  ,  ,  ⊢  | |
 :  ,  :  ,  :  | ||||
| 1 | reference | 10 | ⊢  | |
| 2 | instantiation | 10, 4, 5 | , , , , , , , , , ⊢  | |
: , :  , : | ||||
| 3 | instantiation | 24, 6, 7, 143, 76, 8, 9 | , , , , , , , , , ⊢  | |
 :  ,  :  ,  :  ,  :  ,  :  | ||||
| 4 | instantiation | 10, 11, 12 | , , , , , , , , , ⊢  | |
: , :  , : | ||||
| 5 | instantiation | 24, 13, 14, 122, 59, 15, 16 | , , , , , , , , , ⊢  | |
:  , :  , :  , :  , :  | ||||
| 6 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.nat4 | ||||
| 7 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.nat3 | ||||
| 8 | instantiation | 119 | ⊢  | |
 :  , :  ,  :  | ||||
| 9 | instantiation | 128, 17, 18 | , , , , , , , , ⊢  | |
 :  , :  , :  | ||||
| 10 | axiom | ⊢ | ||
| proveit.logic.equality.equals_transitivity | ||||
| 11 | instantiation | 24, 19, 20, 153, 21, 22, 23 | , , , , , , , , , ⊢  | |
:  , :  , :  , :  , :  | ||||
| 12 | instantiation | 24, 25, 26, 141, 47, 27 | , , , , , , , , , ⊢  | |
:  , :  , :  , :  , :  | ||||
| 13 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.nat5 | ||||
| 14 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.nat2 | ||||
| 15 | instantiation | 148 | ⊢  | |
| : , : | ||||
| 16 | instantiation | 128, 28, 29 | , , , , , , , , ⊢  | |
: , :  , :  | ||||
| 17 | instantiation | 43, 30, 45 | , , , , , , , , ⊢  | |
:  , : | ||||
| 18 | instantiation | 137, 46, 31 | ⊢  | |
 : , :  ,  : | ||||
| 19 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.nat7 | ||||
| 20 | axiom | ⊢  | ||
| proveit.numbers.number_sets.natural_numbers.zero_in_nats | ||||
| 21 | instantiation | 32 | ⊢  | |
: , : , : ,  : ,  : ,  : ,  : | ||||
| 22 | theorem | ⊢  | ||
| proveit.core_expr_types.tuples.tuple_len_0_typical_eq | ||||
| 23 | instantiation | 128, 33, 34 | , , , , , , , , ⊢  | |
: , :  , :  | ||||
| 24 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_pulling_scalar_out_front | ||||
| 25 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.nat6 | ||||
| 26 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.nat1 | ||||
| 27 | instantiation | 128, 35, 36 | , , , , , , , , ⊢  | |
: , :  , :  | ||||
| 28 | instantiation | 43, 37, 45 | , , , , , , , , ⊢  | |
:  , : | ||||
| 29 | instantiation | 137, 46, 38 | ⊢  | |
: , :  , : | ||||
| 30 | instantiation | 128, 39, 40 | , , , , , , , , ⊢  | |
:  , : , :  | ||||
| 31 | instantiation | 57 | ⊢  | |
| : , : , : , : , : , : | ||||
| 32 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.tuple_len_7_typical_eq | ||||
| 33 | instantiation | 43, 41, 45 | , , , , , , , , ⊢  | |
:  , : | ||||
| 34 | instantiation | 137, 46, 42 | ⊢  | |
: , :  , : | ||||
| 35 | instantiation | 43, 44, 45 | , , , , , , , , ⊢  | |
:  , : | ||||
| 36 | instantiation | 137, 46, 47 | ⊢  | |
| : , : , : | ||||
| 37 | instantiation | 128, 48, 49 | , , , , , , , , ⊢  | |
: , : , :  | ||||
| 38 | instantiation | 57 | ⊢  | |
| : , : , : , : , : , : | ||||
| 39 | instantiation | 77, 124, 154, 50, 79 | , , , , , , , , ⊢  | |
 :  ,  :  , :  ,  : | ||||
| 40 | instantiation | 137, 64, 51 | ⊢  | |
: , :  , : | ||||
| 41 | instantiation | 128, 52, 53 | , , , , , , , , ⊢  | |
: , : , :  | ||||
| 42 | instantiation | 57 | ⊢  | |
| : , : , : , : , : , : | ||||
| 43 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_complex_left_closure | ||||
| 44 | instantiation | 128, 54, 55 | , , , , , , , , ⊢  | |
: , : , :  | ||||
| 45 | instantiation | 56, 161, 162 | , ⊢  | |
:  ,  : | ||||
| 46 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.posnat6 | ||||
| 47 | instantiation | 57 | ⊢  | |
| : , : , : , : , : , : | ||||
| 48 | instantiation | 77, 124, 154, 58, 79 | , , , , , , , , ⊢  | |
: , : , :  , : | ||||
| 49 | instantiation | 137, 64, 59 | ⊢  | |
| : , : , : | ||||
| 50 | instantiation | 146, 124, 154, 60 | , , , , , , , ⊢  | |
| : , : , : | ||||
| 51 | instantiation | 72 | ⊢  | |
| : , : , : , : , : | ||||
| 52 | instantiation | 140, 61, 154, 141 | , , , , , , , , ⊢  | |
:  , : , : | ||||
| 53 | instantiation | 137, 64, 62 | ⊢  | |
: , :  , : | ||||
| 54 | instantiation | 77, 124, 154, 63, 79 | , , , , , , , , ⊢  | |
: , : , :  , : | ||||
| 55 | instantiation | 137, 64, 65 | ⊢  | |
: , :  , : | ||||
| 56 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_bra_in_QmultCodomain | ||||
| 57 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.tuple_len_6_typical_eq | ||||
| 58 | instantiation | 146, 124, 154, 66 | , , , , , , , ⊢  | |
| : , : , : | ||||
| 59 | instantiation | 72 | ⊢  | |
| : , : , : , : , : | ||||
| 60 | instantiation | 128, 67, 68 | , , , , , , , ⊢  | |
:  , : , :  | ||||
| 61 | instantiation | 128, 69, 70 | , , , , , , , ⊢  | |
: , : , :  | ||||
| 62 | instantiation | 72 | ⊢  | |
| : , : , : , : , : | ||||
| 63 | instantiation | 146, 124, 154, 71 | , , , , , , , ⊢  | |
| : , : , : | ||||
| 64 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.posnat5 | ||||
| 65 | instantiation | 72 | ⊢  | |
| : , : , : , : , : | ||||
| 66 | instantiation | 128, 73, 74 | , , , , , , , ⊢  | |
: , : , :  | ||||
| 67 | instantiation | 123, 124, 161, 154, 75, 126 | , , , , , , , ⊢  | |
: , : ,  : , :  , : | ||||
| 68 | instantiation | 137, 88, 76 | ⊢  | |
| : , : , : | ||||
| 69 | instantiation | 77, 124, 154, 78, 79 | , , , , , , , ⊢  | |
: , : , :  , : | ||||
| 70 | instantiation | 137, 88, 80 | ⊢  | |
: , :  , : | ||||
| 71 | instantiation | 128, 81, 82 | , , , , , , , ⊢  | |
: , : , :  | ||||
| 72 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.tuple_len_5_typical_eq | ||||
| 73 | instantiation | 123, 124, 161, 154, 83, 126 | , , , , , , , ⊢  | |
: , : , : , :  , : | ||||
| 74 | instantiation | 137, 88, 84 | ⊢  | |
: , :  , : | ||||
| 75 | instantiation | 146, 161, 154, 85 | , , , , ⊢  | |
| : , : , : | ||||
| 76 | instantiation | 96 | ⊢  | |
| : , : , : , : | ||||
| 77 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_op_ket_is_ket | ||||
| 78 | instantiation | 146, 124, 154, 86 | , , , , , , ⊢  | |
| : , : , : | ||||
| 79 | assumption | ⊢ | ||
| 80 | instantiation | 96 | ⊢  | |
| : , : , : , : | ||||
| 81 | instantiation | 142, 122, 124, 154, 87 | , , , , , , , ⊢  | |
: , : , : , :  | ||||
| 82 | instantiation | 137, 88, 89 | ⊢  | |
: , :  , : | ||||
| 83 | instantiation | 146, 161, 154, 90 | , , , , ⊢  | |
| : , : , : | ||||
| 84 | instantiation | 96 | ⊢  | |
| : , : , : , : | ||||
| 85 | instantiation | 128, 91, 92 | , , , , ⊢  | |
:  , : , :  | ||||
| 86 | instantiation | 128, 93, 94 | , , , , , , ⊢  | |
: , : , :  | ||||
| 87 | instantiation | 146, 124, 154, 95 | , , , , , , ⊢  | |
| : , : , : | ||||
| 88 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.posnat4 | ||||
| 89 | instantiation | 96 | ⊢  | |
| : , : , : , : | ||||
| 90 | instantiation | 128, 97, 98 | , , , , ⊢  | |
: , : , :  | ||||
| 91 | instantiation | 152, 99, 161, 154, 155 | , , , , ⊢  | |
:  , : , : , : | ||||
| 92 | instantiation | 137, 111, 100 | ⊢  | |
: , :  , : | ||||
| 93 | instantiation | 142, 122, 124, 154, 101 | , , , , , , ⊢  | |
: , : , : , :  | ||||
| 94 | instantiation | 137, 111, 102 | ⊢  | |
: , :  , : | ||||
| 95 | instantiation | 128, 103, 104 | , , , , , , ⊢  | |
: , : , :  | ||||
| 96 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.tuple_len_4_typical_eq | ||||
| 97 | instantiation | 142, 143, 161, 154, 105 | , , , , ⊢  | |
: , : , : , :  | ||||
| 98 | instantiation | 137, 111, 106 | ⊢  | |
: , :  , : | ||||
| 99 | instantiation | 128, 107, 108 | , , ⊢  | |
: , : , :  | ||||
| 100 | instantiation | 119 | ⊢  | |
| : , : , : | ||||
| 101 | instantiation | 146, 124, 154, 109 | , , , , , ⊢  | |
| : , : , : | ||||
| 102 | instantiation | 119 | ⊢  | |
| : , : , : | ||||
| 103 | instantiation | 123, 124, 161, 154, 110, 126 | , , , , , , ⊢  | |
: , : , : , :  , : | ||||
| 104 | instantiation | 137, 111, 112 | ⊢  | |
: , :  , : | ||||
| 105 | instantiation | 146, 161, 154, 113 | , , , ⊢  | |
| : , : , : | ||||
| 106 | instantiation | 119 | ⊢  | |
| : , : , : | ||||
| 107 | instantiation | 140, 114, 154, 153 | , , ⊢  | |
:  , : , : | ||||
| 108 | instantiation | 137, 138, 115 | ⊢  | |
: , :  , : | ||||
| 109 | instantiation | 128, 116, 117 | , , , , , ⊢  | |
: , : , :  | ||||
| 110 | instantiation | 146, 161, 154, 118 | , , , ⊢  | |
| : , : , : | ||||
| 111 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.posnat3 | ||||
| 112 | instantiation | 119 | ⊢  | |
| : , : , : | ||||
| 113 | instantiation | 128, 120, 121 | , , , ⊢  | |
: , : , :  | ||||
| 114 | instantiation | 140, 122, 154, 141 | , ⊢  | |
| : , : , : | ||||
| 115 | instantiation | 148 | ⊢  | |
| : , : | ||||
| 116 | instantiation | 123, 124, 161, 154, 125, 126 | , , , , , ⊢  | |
: , : , : , :  , : | ||||
| 117 | instantiation | 137, 138, 127 | ⊢  | |
: , :  , : | ||||
| 118 | instantiation | 128, 129, 130 | , , , ⊢  | |
: , : , :  | ||||
| 119 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.tuple_len_3_typical_eq | ||||
| 120 | instantiation | 152, 131, 161, 154, 155 | , , , ⊢  | |
:  , : , : , : | ||||
| 121 | instantiation | 137, 138, 132 | ⊢  | |
: , :  , : | ||||
| 122 | assumption | ⊢ | ||
| 123 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_op_op_is_op | ||||
| 124 | instantiation | 166, 157 | ⊢  | |
| : | ||||
| 125 | instantiation | 146, 161, 154, 133 | , , ⊢  | |
| : , : , : | ||||
| 126 | instantiation | 134, 167, 157, 135 | , , , ⊢  | |
: , : ,  : | ||||
| 127 | instantiation | 148 | ⊢  | |
| : , : | ||||
| 128 | theorem | ⊢  | ||
| proveit.logic.equality.sub_left_side_into | ||||
| 129 | instantiation | 142, 143, 161, 154, 136 | , , , ⊢  | |
: , : , : , :  | ||||
| 130 | instantiation | 137, 138, 139 | ⊢  | |
: , :  , : | ||||
| 131 | instantiation | 140, 141, 154, 153 | , ⊢  | |
| : , : , : | ||||
| 132 | instantiation | 148 | ⊢  | |
| : , : | ||||
| 133 | instantiation | 142, 143, 161, 154, 155 | , , ⊢  | |
| : , : , : , : | ||||
| 134 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_matrix_is_linmap | ||||
| 135 | modus ponens | 144, 145 | , , , ⊢  | |
| 136 | instantiation | 146, 161, 154, 147 | , , ⊢  | |
| : , : , : | ||||
| 137 | axiom | ⊢  | ||
| proveit.physics.quantum.algebra.multi_qmult_def | ||||
| 138 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.posnat2 | ||||
| 139 | instantiation | 148 | ⊢  | |
| : , : | ||||
| 140 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_complex_ket_closure | ||||
| 141 | assumption | ⊢ | ||
| 142 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_op_complex_closure | ||||
| 143 | assumption | ⊢ | ||
| 144 | instantiation | 149, 165, 150, 151 | , , ⊢  | |
 : ,  : ,  : ,  : ,  :  , :  | ||||
| 145 | assumption | ⊢ | ||
| 146 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_op_is_linmap | ||||
| 147 | instantiation | 152, 153, 161, 154, 155 | , , ⊢  | |
| : , : , : , : | ||||
| 148 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.tuple_len_2_typical_eq | ||||
| 149 | theorem | ⊢  | ||
| proveit.linear_algebra.addition.summation_closure | ||||
| 150 | instantiation | 156, 167, 157 | , ⊢  | |
| : , : | ||||
| 151 | instantiation | 158, 159 | ⊢  | |
 : ,  :  | ||||
| 152 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_complex_op_closure | ||||
| 153 | assumption | ⊢ | ||
| 154 | theorem | ⊢  | ||
| proveit.linear_algebra.inner_products.complex_set_is_hilbert_space | ||||
| 155 | instantiation | 160, 161, 162 | , ⊢  | |
| : , : | ||||
| 156 | theorem | ⊢  | ||
| proveit.linear_algebra.matrices.complex_matrix_space_is_vec_space | ||||
| 157 | assumption | ⊢ | ||
| 158 | theorem | ⊢  | ||
| proveit.core_expr_types.tuples.range_from1_len_typical_eq | ||||
| 159 | instantiation | 163, 164, 165 | ⊢  | |
:  , :  , : | ||||
| 160 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_bra_is_linmap | ||||
| 161 | instantiation | 166, 167 | ⊢  | |
| : | ||||
| 162 | assumption | ⊢ | ||
| 163 | theorem | ⊢  | ||
| proveit.logic.sets.inclusion.superset_membership_from_proper_subset | ||||
| 164 | theorem | ⊢  | ||
| proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat | ||||
| 165 | assumption | ⊢ | ||
| 166 | theorem | ⊢  | ||
| proveit.linear_algebra.inner_products.complex_vec_set_is_hilbert_space | ||||
| 167 | assumption | ⊢ | ||