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