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 | 32 | ⊢  | |
| 2 | instantiation | 4, 5 | , , , ⊢  | |
 :  , :  , :  | ||||
| 3 | instantiation | 43, 6, 7, 8 | , , , , , , ⊢  | |
 : ,  :  ,  :  ,  : | ||||
| 4 | axiom | ⊢  | ||
| proveit.logic.equality.substitution | ||||
| 5 | instantiation | 32, 9, 10 | , , , ⊢  | |
: , :  , : | ||||
| 6 | instantiation | 11, 126, 79, 78, 41, 12 | , , , , , , ⊢  | |
 :  ,  :  ,  :  ,  :  ,  :  ,  :  | ||||
| 7 | instantiation | 39, 78, 40, 119, 41, 13 | , , , , , , ⊢  | |
: ,  : , : , :  , :  , : | ||||
| 8 | instantiation | 32, 14, 15 | , , , , , , ⊢  | |
: , :  , : | ||||
| 9 | instantiation | 62, 79, 77, 126, 80, 16 | , , , ⊢  | |
: , :  , : , : , :  | ||||
| 10 | instantiation | 47, 40, 126, 16 | , , , ⊢  | |
 : , : , : | ||||
| 11 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.scalar_mult_absorption | ||||
| 12 | instantiation | 109, 17, 18 | , , , , , ⊢  | |
 :  , :  , :  | ||||
| 13 | instantiation | 109, 19, 20 | , , , , , , ⊢  | |
: , : , :  | ||||
| 14 | instantiation | 32, 21, 22 | , , , , , , ⊢  | |
: , :  , : | ||||
| 15 | instantiation | 64, 90, 78, 23, 132, 126, 128, 66, 67, 24* | , , , , , , ⊢  | |
:  , : , :  , :  | ||||
| 16 | instantiation | 25, 138, 139, 140 | , , ⊢  | |
: , : ,  : | ||||
| 17 | instantiation | 54, 26, 29 | , , , , , ⊢  | |
: , :  , : | ||||
| 18 | instantiation | 43, 27, 28, 29 | , , , , , ⊢  | |
: , :  , : , : | ||||
| 19 | instantiation | 82, 128, 30 | , , , , , , ⊢  | |
:  , :  | ||||
| 20 | instantiation | 121, 31, 49 | ⊢  | |
:  , :  , : | ||||
| 21 | instantiation | 32, 33, 34 | , , , , , , ⊢  | |
: , :  , : | ||||
| 22 | instantiation | 62, 78, 132, 113, 66, 35 | , , , , , , ⊢  | |
: , : , : , :  , : | ||||
| 23 | instantiation | 100 | ⊢  | |
| : , : , : | ||||
| 24 | instantiation | 47, 122, 36, 66, 67 | , , , , , , ⊢  | |
: , :  , : | ||||
| 25 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_matrix_in_QmultCodomain | ||||
| 26 | instantiation | 109, 37, 38 | , , , , , ⊢  | |
: , : , :  | ||||
| 27 | instantiation | 39, 79, 40, 78, 41, 42 | , , , , , ⊢  | |
| : , : , : , : , : , : | ||||
| 28 | instantiation | 43, 44, 45, 46 | , , , , , ⊢  | |
: , :  , :  , : | ||||
| 29 | instantiation | 47, 122, 93, 66, 67 | , , , , , ⊢  | |
: , :  , : | ||||
| 30 | instantiation | 94, 50 | , , , , , ⊢  | |
| : | ||||
| 31 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.posnat4 | ||||
| 32 | axiom | ⊢ | ||
| proveit.logic.equality.equals_transitivity | ||||
| 33 | instantiation | 62, 48, 77, 128, 49, 80, 50 | , , , , , , ⊢  | |
| : , : , : , : , : | ||||
| 34 | instantiation | 62, 78, 126, 123, 66, 51 | , , , , , , ⊢  | |
: , : , : , :  , : | ||||
| 35 | instantiation | 109, 52, 53 | , , , , , ⊢  | |
: , :  , :  | ||||
| 36 | instantiation | 54, 55, 56 | , , ⊢  | |
:  , :  , : | ||||
| 37 | instantiation | 114, 133, 134, 57, 116 | , , , , , ⊢  | |
 :  ,  :  , :  ,  :  | ||||
| 38 | instantiation | 121, 122, 58 | ⊢  | |
: , :  , : | ||||
| 39 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_disassociation | ||||
| 40 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.posnat1 | ||||
| 41 | instantiation | 130 | ⊢  | |
| : , : | ||||
| 42 | instantiation | 109, 59, 60 | , , , , , ⊢  | |
: , : , :  | ||||
| 43 | theorem | ⊢  | ||
| proveit.logic.equality.four_chain_transitivity | ||||
| 44 | instantiation | 62, 61, 77, 128, 84, 80, 95 | , , , , , ⊢  | |
: , : , : , :  , : | ||||
| 45 | instantiation | 62, 79, 78, 132, 66, 63 | , , , , , ⊢  | |
: , : , : , :  , : | ||||
| 46 | instantiation | 64, 122, 78, 65, 132, 128, 66, 67 | , , , , , ⊢  | |
: , : , :  , : | ||||
| 47 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.scalar_mult_factorization | ||||
| 48 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.nat4 | ||||
| 49 | instantiation | 68 | ⊢  | |
| : , : , : , : | ||||
| 50 | instantiation | 109, 69, 70 | , , , , , ⊢  | |
: , : , :  | ||||
| 51 | instantiation | 109, 71, 72 | , , , , , ⊢  | |
: , : , :  | ||||
| 52 | instantiation | 114, 133, 134, 73, 116 | , , , , , ⊢  | |
: , : , :  , : | ||||
| 53 | instantiation | 121, 90, 74 | ⊢  | |
: , :  , : | ||||
| 54 | theorem | ⊢  | ||
| proveit.logic.equality.sub_right_side_into | ||||
| 55 | instantiation | 103, 75, 128 | , , ⊢  | |
:  , : | ||||
| 56 | instantiation | 76, 77, 78, 79, 80, 119, 132, 126, 128 | , , ⊢  | |
 : , : ,  : , : , : , : | ||||
| 57 | instantiation | 124, 133, 134, 81 | , , , , ⊢  | |
| : , : , : | ||||
| 58 | instantiation | 130 | ⊢  | |
| : , : | ||||
| 59 | instantiation | 82, 128, 83 | , , , , , ⊢  | |
: , :  | ||||
| 60 | instantiation | 121, 90, 84 | ⊢  | |
| : , : , : | ||||
| 61 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.nat3 | ||||
| 62 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_pulling_scalar_out_front | ||||
| 63 | instantiation | 109, 85, 86 | , , , , ⊢  | |
: , :  , :  | ||||
| 64 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_scalar_association | ||||
| 65 | instantiation | 130 | ⊢  | |
| : , : | ||||
| 66 | instantiation | 130 | ⊢  | |
| : , : | ||||
| 67 | instantiation | 114, 133, 134, 135, 116 | , , , ⊢  | |
| : , : , : , : | ||||
| 68 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.tuple_len_4_typical_eq | ||||
| 69 | instantiation | 114, 133, 134, 87, 116 | , , , , , ⊢  | |
: , : , :  , : | ||||
| 70 | instantiation | 121, 90, 88 | ⊢  | |
: , :  , : | ||||
| 71 | instantiation | 114, 133, 134, 89, 116 | , , , , , ⊢  | |
: , : , :  , : | ||||
| 72 | instantiation | 121, 90, 91 | ⊢  | |
: , :  , : | ||||
| 73 | instantiation | 124, 133, 134, 92 | , , , , ⊢  | |
| : , : , : | ||||
| 74 | instantiation | 100 | ⊢  | |
| : , : , : | ||||
| 75 | instantiation | 103, 132, 126 | , ⊢  | |
| : , : | ||||
| 76 | theorem | ⊢  | ||
| proveit.numbers.multiplication.disassociation | ||||
| 77 | axiom | ⊢  | ||
| proveit.numbers.number_sets.natural_numbers.zero_in_nats | ||||
| 78 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.nat2 | ||||
| 79 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.nat1 | ||||
| 80 | theorem | ⊢  | ||
| proveit.core_expr_types.tuples.tuple_len_0_typical_eq | ||||
| 81 | instantiation | 131, 93, 133, 134, 135 | , , , , ⊢  | |
| : , : , : , : | ||||
| 82 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_complex_right_closure | ||||
| 83 | instantiation | 94, 95 | , , , , ⊢  | |
| : | ||||
| 84 | instantiation | 100 | ⊢  | |
| : , : , : | ||||
| 85 | instantiation | 114, 133, 134, 96, 116 | , , , , ⊢  | |
: , : , :  , : | ||||
| 86 | instantiation | 121, 122, 97 | ⊢  | |
: , :  , : | ||||
| 87 | instantiation | 124, 133, 134, 98 | , , , , ⊢  | |
| : , : , : | ||||
| 88 | instantiation | 100 | ⊢  | |
| : , : , : | ||||
| 89 | instantiation | 124, 133, 134, 99 | , , , , ⊢  | |
| : , : , : | ||||
| 90 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.posnat3 | ||||
| 91 | instantiation | 100 | ⊢  | |
| : , : , : | ||||
| 92 | instantiation | 109, 101, 102 | , , , , ⊢  | |
:  , : , :  | ||||
| 93 | instantiation | 103, 132, 128 | , ⊢  | |
| : , : | ||||
| 94 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_nested_closure | ||||
| 95 | instantiation | 109, 104, 105 | , , , , ⊢  | |
: , : , :  | ||||
| 96 | instantiation | 124, 133, 134, 106 | , , , ⊢  | |
| : , : , : | ||||
| 97 | instantiation | 130 | ⊢  | |
| : , : | ||||
| 98 | instantiation | 109, 107, 108 | , , , , ⊢  | |
: , : , :  | ||||
| 99 | instantiation | 109, 110, 111 | , , , , ⊢  | |
: , : , :  | ||||
| 100 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.tuple_len_3_typical_eq | ||||
| 101 | instantiation | 131, 112, 133, 134, 135 | , , , , ⊢  | |
:  , : , : , : | ||||
| 102 | instantiation | 121, 122, 113 | ⊢  | |
| : , : , : | ||||
| 103 | theorem | ⊢  | ||
| proveit.numbers.multiplication.mult_complex_closure_bin | ||||
| 104 | instantiation | 114, 133, 134, 115, 116 | , , , , ⊢  | |
: , : , :  , : | ||||
| 105 | instantiation | 121, 122, 117 | ⊢  | |
: , :  , : | ||||
| 106 | instantiation | 131, 128, 133, 134, 135 | , , , ⊢  | |
| : , : , : , : | ||||
| 107 | instantiation | 131, 118, 133, 134, 135 | , , , , ⊢  | |
:  , : , : , : | ||||
| 108 | instantiation | 121, 122, 119 | ⊢  | |
| : , : , : | ||||
| 109 | theorem | ⊢  | ||
| proveit.logic.equality.sub_left_side_into | ||||
| 110 | instantiation | 131, 120, 133, 134, 135 | , , , , ⊢  | |
:  , : , : , : | ||||
| 111 | instantiation | 121, 122, 123 | ⊢  | |
| : , : , : | ||||
| 112 | instantiation | 127, 126, 129, 128 | , ⊢  | |
: , :  , : | ||||
| 113 | instantiation | 130 | ⊢  | |
| : , : | ||||
| 114 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_op_ket_in_QmultCodomain | ||||
| 115 | instantiation | 124, 133, 134, 125 | , , , ⊢  | |
| : , : , : | ||||
| 116 | assumption | ⊢ | ||
| 117 | instantiation | 130 | ⊢  | |
| : , : | ||||
| 118 | instantiation | 127, 132, 129, 126 | , ⊢  | |
| : , : , : | ||||
| 119 | instantiation | 130 | ⊢  | |
| : , : | ||||
| 120 | instantiation | 127, 128, 129, 132 | , ⊢  | |
| : , : , : | ||||
| 121 | axiom | ⊢  | ||
| proveit.physics.quantum.algebra.multi_qmult_def | ||||
| 122 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.posnat2 | ||||
| 123 | instantiation | 130 | ⊢  | |
| : , : | ||||
| 124 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_op_is_linmap | ||||
| 125 | instantiation | 131, 132, 133, 134, 135 | , , , ⊢  | |
| : , : , : , : | ||||
| 126 | assumption | ⊢ | ||
| 127 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_complex_ket_closure | ||||
| 128 | assumption | ⊢ | ||
| 129 | theorem | ⊢  | ||
| proveit.linear_algebra.inner_products.complex_set_is_hilbert_space | ||||
| 130 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.tuple_len_2_typical_eq | ||||
| 131 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_complex_op_closure | ||||
| 132 | assumption | ⊢ | ||
| 133 | instantiation | 136, 139 | ⊢  | |
| : | ||||
| 134 | instantiation | 136, 138 | ⊢  | |
| : | ||||
| 135 | instantiation | 137, 138, 139, 140 | , , ⊢  | |
| : , : , : | ||||
| 136 | theorem | ⊢  | ||
| proveit.linear_algebra.inner_products.complex_vec_set_is_hilbert_space | ||||
| 137 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_matrix_is_linmap | ||||
| 138 | assumption | ⊢ | ||
| 139 | assumption | ⊢ | ||
| 140 | assumption | ⊢ | ||
| *equality replacement requirements | ||||