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 | 18 | ⊢  | |
| 2 | instantiation | 18, 4, 5 | , , , , , , , , ⊢  | |
: , :  , : | ||||
| 3 | instantiation | 6, 103, 12, 7, 159, 137, 157, 164, 8, 9 | , , , , , , , , ⊢  | |
 :  ,  : ,  :  ,  :  | ||||
| 4 | instantiation | 18, 10, 11 | , , , , , , , , ⊢  | |
: , :  , : | ||||
| 5 | instantiation | 34, 12, 13, 159, 91, 14, 15 | , , , , , , , , ⊢  | |
: ,  :  , :  , :  ,  :  | ||||
| 6 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_scalar_association | ||||
| 7 | instantiation | 111 | ⊢  | |
 : , :  ,  :  ,  :  | ||||
| 8 | instantiation | 111 | ⊢  | |
:  , : , :  , : | ||||
| 9 | instantiation | 143, 16, 17 | , , , , ⊢  | |
 :  , :  , :  | ||||
| 10 | instantiation | 18, 19, 20 | , , , , , , , , ⊢  | |
: , :  , : | ||||
| 11 | instantiation | 34, 21, 22, 137, 74, 23, 24 | , , , , , , , , ⊢  | |
:  , :  , : , :  , :  | ||||
| 12 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.nat4 | ||||
| 13 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.nat3 | ||||
| 14 | instantiation | 134 | ⊢  | |
| : , : , : | ||||
| 15 | instantiation | 143, 25, 26 | , , , , , , , ⊢  | |
: , :  , :  | ||||
| 16 | instantiation | 57, 27, 59 | , , , , ⊢  | |
:  , : | ||||
| 17 | instantiation | 153, 126, 28 | ⊢  | |
 : , :  ,  : | ||||
| 18 | axiom | ⊢ | ||
| proveit.logic.equality.equals_transitivity | ||||
| 19 | instantiation | 34, 29, 30, 164, 31, 32, 33 | , , , , , , , , ⊢  | |
:  , :  , : , :  , :  | ||||
| 20 | instantiation | 34, 35, 36, 157, 61, 37 | , , , , , , , , ⊢  | |
:  , :  , : , :  , :  | ||||
| 21 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.nat5 | ||||
| 22 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.nat2 | ||||
| 23 | instantiation | 162 | ⊢  | |
| : , : | ||||
| 24 | instantiation | 143, 38, 39 | , , , , , , , ⊢  | |
: , :  , :  | ||||
| 25 | instantiation | 57, 40, 59 | , , , , , , , ⊢  | |
:  , : | ||||
| 26 | instantiation | 153, 60, 41 | ⊢  | |
: , :  , : | ||||
| 27 | instantiation | 143, 42, 43 | , , , , ⊢  | |
:  , : , :  | ||||
| 28 | instantiation | 134 | ⊢  | |
| : , : , : | ||||
| 29 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.nat7 | ||||
| 30 | axiom | ⊢  | ||
| proveit.numbers.number_sets.natural_numbers.zero_in_nats | ||||
| 31 | instantiation | 44 | ⊢  | |
: , : , : , : ,  : ,  : ,  : | ||||
| 32 | theorem | ⊢  | ||
| proveit.core_expr_types.tuples.tuple_len_0_typical_eq | ||||
| 33 | instantiation | 143, 45, 46 | , , , , , , , ⊢  | |
: , :  , :  | ||||
| 34 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_pulling_scalar_out_front | ||||
| 35 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.nat6 | ||||
| 36 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.nat1 | ||||
| 37 | instantiation | 143, 47, 48 | , , , , , , , ⊢  | |
: , :  , :  | ||||
| 38 | instantiation | 57, 49, 59 | , , , , , , , ⊢  | |
:  , : | ||||
| 39 | instantiation | 153, 60, 50 | ⊢  | |
: , :  , : | ||||
| 40 | instantiation | 143, 51, 52 | , , , , , , , ⊢  | |
: , : , :  | ||||
| 41 | instantiation | 72 | ⊢  | |
| : , : , : , : , : , : | ||||
| 42 | instantiation | 92, 139, 165, 53, 94 | , , , , ⊢  | |
 :  ,  :  , :  ,  : | ||||
| 43 | instantiation | 153, 154, 54 | ⊢  | |
: , :  , : | ||||
| 44 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.tuple_len_7_typical_eq | ||||
| 45 | instantiation | 57, 55, 59 | , , , , , , , ⊢  | |
:  , : | ||||
| 46 | instantiation | 153, 60, 56 | ⊢  | |
: , :  , : | ||||
| 47 | instantiation | 57, 58, 59 | , , , , , , , ⊢  | |
:  , : | ||||
| 48 | instantiation | 153, 60, 61 | ⊢  | |
| : , : , : | ||||
| 49 | instantiation | 143, 62, 63 | , , , , , , , ⊢  | |
: , : , :  | ||||
| 50 | instantiation | 72 | ⊢  | |
| : , : , : , : , : , : | ||||
| 51 | instantiation | 92, 139, 165, 64, 94 | , , , , , , , ⊢  | |
: , : , :  , : | ||||
| 52 | instantiation | 153, 79, 65 | ⊢  | |
: , :  , : | ||||
| 53 | instantiation | 160, 139, 165, 66 | , , , ⊢  | |
| : , : , : | ||||
| 54 | instantiation | 162 | ⊢  | |
| : , : | ||||
| 55 | instantiation | 143, 67, 68 | , , , , , , , ⊢  | |
: , : , :  | ||||
| 56 | instantiation | 72 | ⊢  | |
| : , : , : , : , : , : | ||||
| 57 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_complex_left_closure | ||||
| 58 | instantiation | 143, 69, 70 | , , , , , , , ⊢  | |
: , : , :  | ||||
| 59 | instantiation | 71, 168, 169 | , ⊢  | |
:  ,  : | ||||
| 60 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.posnat6 | ||||
| 61 | instantiation | 72 | ⊢  | |
| : , : , : , : , : , : | ||||
| 62 | instantiation | 92, 139, 165, 73, 94 | , , , , , , , ⊢  | |
: , : , :  , : | ||||
| 63 | instantiation | 153, 79, 74 | ⊢  | |
| : , : , : | ||||
| 64 | instantiation | 160, 139, 165, 75 | , , , , , , ⊢  | |
| : , : , : | ||||
| 65 | instantiation | 87 | ⊢  | |
| : , : , : , : , : | ||||
| 66 | instantiation | 138, 139, 168, 165, 166, 141 | , , , ⊢  | |
: , : ,  : , : , : | ||||
| 67 | instantiation | 156, 76, 165, 157 | , , , , , , , ⊢  | |
:  , : , : | ||||
| 68 | instantiation | 153, 79, 77 | ⊢  | |
: , :  , : | ||||
| 69 | instantiation | 92, 139, 165, 78, 94 | , , , , , , , ⊢  | |
: , : , :  , : | ||||
| 70 | instantiation | 153, 79, 80 | ⊢  | |
: , :  , : | ||||
| 71 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_bra_in_QmultCodomain | ||||
| 72 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.tuple_len_6_typical_eq | ||||
| 73 | instantiation | 160, 139, 165, 81 | , , , , , , ⊢  | |
| : , : , : | ||||
| 74 | instantiation | 87 | ⊢  | |
| : , : , : , : , : | ||||
| 75 | instantiation | 143, 82, 83 | , , , , , , ⊢  | |
:  , : , :  | ||||
| 76 | instantiation | 143, 84, 85 | , , , , , , ⊢  | |
: , : , :  | ||||
| 77 | instantiation | 87 | ⊢  | |
| : , : , : , : , : | ||||
| 78 | instantiation | 160, 139, 165, 86 | , , , , , , ⊢  | |
| : , : , : | ||||
| 79 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.posnat5 | ||||
| 80 | instantiation | 87 | ⊢  | |
| : , : , : , : , : | ||||
| 81 | instantiation | 143, 88, 89 | , , , , , , ⊢  | |
: , : , :  | ||||
| 82 | instantiation | 138, 139, 168, 165, 90, 141 | , , , , , , ⊢  | |
: , : , : , :  , : | ||||
| 83 | instantiation | 153, 103, 91 | ⊢  | |
| : , : , : | ||||
| 84 | instantiation | 92, 139, 165, 93, 94 | , , , , , , ⊢  | |
: , : , :  , : | ||||
| 85 | instantiation | 153, 103, 95 | ⊢  | |
: , :  , : | ||||
| 86 | instantiation | 143, 96, 97 | , , , , , , ⊢  | |
: , : , :  | ||||
| 87 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.tuple_len_5_typical_eq | ||||
| 88 | instantiation | 138, 139, 168, 165, 98, 141 | , , , , , , ⊢  | |
: , : , : , :  , : | ||||
| 89 | instantiation | 153, 103, 99 | ⊢  | |
: , :  , : | ||||
| 90 | instantiation | 160, 168, 165, 100 | , , , , ⊢  | |
| : , : , : | ||||
| 91 | instantiation | 111 | ⊢  | |
| : , : , : , : | ||||
| 92 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_op_ket_is_ket | ||||
| 93 | instantiation | 160, 139, 165, 101 | , , , , , ⊢  | |
| : , : , : | ||||
| 94 | assumption | ⊢ | ||
| 95 | instantiation | 111 | ⊢  | |
| : , : , : , : | ||||
| 96 | instantiation | 158, 137, 139, 165, 102 | , , , , , , ⊢  | |
: , : , : , :  | ||||
| 97 | instantiation | 153, 103, 104 | ⊢  | |
: , :  , : | ||||
| 98 | instantiation | 160, 168, 165, 105 | , , , , ⊢  | |
| : , : , : | ||||
| 99 | instantiation | 111 | ⊢  | |
| : , : , : , : | ||||
| 100 | instantiation | 143, 106, 107 | , , , , ⊢  | |
:  , : , :  | ||||
| 101 | instantiation | 143, 108, 109 | , , , , , ⊢  | |
: , : , :  | ||||
| 102 | instantiation | 160, 139, 165, 110 | , , , , , ⊢  | |
| : , : , : | ||||
| 103 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.posnat4 | ||||
| 104 | instantiation | 111 | ⊢  | |
| : , : , : , : | ||||
| 105 | instantiation | 143, 112, 113 | , , , , ⊢  | |
: , : , :  | ||||
| 106 | instantiation | 163, 114, 168, 165, 166 | , , , , ⊢  | |
:  , : , : , : | ||||
| 107 | instantiation | 153, 126, 115 | ⊢  | |
: , :  , : | ||||
| 108 | instantiation | 158, 137, 139, 165, 116 | , , , , , ⊢  | |
: , : , : , :  | ||||
| 109 | instantiation | 153, 126, 117 | ⊢  | |
: , :  , : | ||||
| 110 | instantiation | 143, 118, 119 | , , , , , ⊢  | |
: , : , :  | ||||
| 111 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.tuple_len_4_typical_eq | ||||
| 112 | instantiation | 158, 159, 168, 165, 120 | , , , , ⊢  | |
: , : , : , :  | ||||
| 113 | instantiation | 153, 126, 121 | ⊢  | |
: , :  , : | ||||
| 114 | instantiation | 143, 122, 123 | , , ⊢  | |
: , : , :  | ||||
| 115 | instantiation | 134 | ⊢  | |
| : , : , : | ||||
| 116 | instantiation | 160, 139, 165, 124 | , , , , ⊢  | |
| : , : , : | ||||
| 117 | instantiation | 134 | ⊢  | |
| : , : , : | ||||
| 118 | instantiation | 138, 139, 168, 165, 125, 141 | , , , , , ⊢  | |
: , : , : , :  , : | ||||
| 119 | instantiation | 153, 126, 127 | ⊢  | |
: , :  , : | ||||
| 120 | instantiation | 160, 168, 165, 128 | , , , ⊢  | |
| : , : , : | ||||
| 121 | instantiation | 134 | ⊢  | |
| : , : , : | ||||
| 122 | instantiation | 156, 129, 165, 164 | , , ⊢  | |
:  , : , : | ||||
| 123 | instantiation | 153, 154, 130 | ⊢  | |
: , :  , : | ||||
| 124 | instantiation | 143, 131, 132 | , , , , ⊢  | |
: , : , :  | ||||
| 125 | instantiation | 160, 168, 165, 133 | , , , ⊢  | |
| : , : , : | ||||
| 126 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.posnat3 | ||||
| 127 | instantiation | 134 | ⊢  | |
| : , : , : | ||||
| 128 | instantiation | 143, 135, 136 | , , , ⊢  | |
: , : , :  | ||||
| 129 | instantiation | 156, 137, 165, 157 | , ⊢  | |
| : , : , : | ||||
| 130 | instantiation | 162 | ⊢  | |
| : , : | ||||
| 131 | instantiation | 138, 139, 168, 165, 140, 141 | , , , , ⊢  | |
: , : , : , :  , : | ||||
| 132 | instantiation | 153, 154, 142 | ⊢  | |
: , :  , : | ||||
| 133 | instantiation | 143, 144, 145 | , , , ⊢  | |
: , : , :  | ||||
| 134 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.tuple_len_3_typical_eq | ||||
| 135 | instantiation | 163, 146, 168, 165, 166 | , , , ⊢  | |
:  , : , : , : | ||||
| 136 | instantiation | 153, 154, 147 | ⊢  | |
: , :  , : | ||||
| 137 | assumption | ⊢ | ||
| 138 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_op_op_is_op | ||||
| 139 | instantiation | 170, 150 | ⊢  | |
| : | ||||
| 140 | instantiation | 160, 168, 165, 148 | , , ⊢  | |
| : , : , : | ||||
| 141 | instantiation | 149, 171, 150, 151 | , , ⊢  | |
: , : ,  : | ||||
| 142 | instantiation | 162 | ⊢  | |
| : , : | ||||
| 143 | theorem | ⊢  | ||
| proveit.logic.equality.sub_left_side_into | ||||
| 144 | instantiation | 158, 159, 168, 165, 152 | , , , ⊢  | |
: , : , : , :  | ||||
| 145 | instantiation | 153, 154, 155 | ⊢  | |
: , :  , : | ||||
| 146 | instantiation | 156, 157, 165, 164 | , ⊢  | |
| : , : , : | ||||
| 147 | instantiation | 162 | ⊢  | |
| : , : | ||||
| 148 | instantiation | 158, 159, 168, 165, 166 | , , ⊢  | |
| : , : , : , : | ||||
| 149 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_matrix_is_linmap | ||||
| 150 | assumption | ⊢ | ||
| 151 | assumption | ⊢ | ||
| 152 | instantiation | 160, 168, 165, 161 | , , ⊢  | |
| : , : , : | ||||
| 153 | axiom | ⊢  | ||
| proveit.physics.quantum.algebra.multi_qmult_def | ||||
| 154 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.posnat2 | ||||
| 155 | instantiation | 162 | ⊢  | |
| : , : | ||||
| 156 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_complex_ket_closure | ||||
| 157 | assumption | ⊢ | ||
| 158 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_op_complex_closure | ||||
| 159 | assumption | ⊢ | ||
| 160 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_op_is_linmap | ||||
| 161 | instantiation | 163, 164, 168, 165, 166 | , , ⊢  | |
| : , : , : , : | ||||
| 162 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.tuple_len_2_typical_eq | ||||
| 163 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_complex_op_closure | ||||
| 164 | assumption | ⊢ | ||
| 165 | theorem | ⊢  | ||
| proveit.linear_algebra.inner_products.complex_set_is_hilbert_space | ||||
| 166 | instantiation | 167, 168, 169 | , ⊢  | |
| : , : | ||||
| 167 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_bra_is_linmap | ||||
| 168 | instantiation | 170, 171 | ⊢  | |
| : | ||||
| 169 | assumption | ⊢ | ||
| 170 | theorem | ⊢  | ||
| proveit.linear_algebra.inner_products.complex_vec_set_is_hilbert_space | ||||
| 171 | assumption | ⊢ | ||