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, 4 |  ,  ,  ,  ,  ,  ,  ⊢  | |
 :  ,  :  ,  :  ,  :  | ||||
| 1 | reference | 35 | ⊢  | |
| 2 | instantiation | 5, 118, 71, 70, 33, 6 | , , , , , , ⊢  | |
 :  ,  :  ,  :  ,  :  ,  :  ,  :  | ||||
| 3 | instantiation | 31, 70, 32, 111, 33, 7 | , , , , , , ⊢  | |
: ,  : , : , :  , :  , : | ||||
| 4 | instantiation | 24, 8, 9 | , , , , , , ⊢  | |
 : ,  :  ,  : | ||||
| 5 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.scalar_mult_absorption | ||||
| 6 | instantiation | 101, 10, 11 | , , , , , ⊢  | |
 :  , :  , :  | ||||
| 7 | instantiation | 101, 12, 13 | , , , , , , ⊢  | |
: , : , :  | ||||
| 8 | instantiation | 24, 14, 15 | , , , , , , ⊢  | |
: , :  , : | ||||
| 9 | instantiation | 56, 82, 70, 16, 124, 118, 120, 58, 59, 17* | , , , , , , ⊢  | |
 :  , : , :  , :  | ||||
| 10 | instantiation | 46, 18, 21 | , , , , , ⊢  | |
: , :  , : | ||||
| 11 | instantiation | 35, 19, 20, 21 | , , , , , ⊢  | |
: , :  , : , : | ||||
| 12 | instantiation | 74, 120, 22 | , , , , , , ⊢  | |
:  , :  | ||||
| 13 | instantiation | 113, 23, 41 | ⊢  | |
:  , :  , : | ||||
| 14 | instantiation | 24, 25, 26 | , , , , , , ⊢  | |
: , :  , : | ||||
| 15 | instantiation | 54, 70, 124, 105, 58, 27 | , , , , , , ⊢  | |
: , : , : , :  , : | ||||
| 16 | instantiation | 92 | ⊢  | |
| : , : , : | ||||
| 17 | instantiation | 39, 114, 28, 58, 59 | , , , , , , ⊢  | |
: , :  , : | ||||
| 18 | instantiation | 101, 29, 30 | , , , , , ⊢  | |
: , : , :  | ||||
| 19 | instantiation | 31, 71, 32, 70, 33, 34 | , , , , , ⊢  | |
| : , : , : , : , : , : | ||||
| 20 | instantiation | 35, 36, 37, 38 | , , , , , ⊢  | |
: , :  , :  , : | ||||
| 21 | instantiation | 39, 114, 85, 58, 59 | , , , , , ⊢  | |
: , :  , : | ||||
| 22 | instantiation | 86, 42 | , , , , , ⊢  | |
| : | ||||
| 23 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.posnat4 | ||||
| 24 | axiom | ⊢  | ||
| proveit.logic.equality.equals_transitivity | ||||
| 25 | instantiation | 54, 40, 69, 120, 41, 72, 42 | , , , , , , ⊢  | |
: , :  , : , : , :  | ||||
| 26 | instantiation | 54, 70, 118, 115, 58, 43 | , , , , , , ⊢  | |
: , : , : , :  , : | ||||
| 27 | instantiation | 101, 44, 45 | , , , , , ⊢  | |
: , :  , :  | ||||
| 28 | instantiation | 46, 47, 48 | , , ⊢  | |
:  , :  , : | ||||
| 29 | instantiation | 106, 125, 126, 49, 108 | , , , , , ⊢  | |
 :  ,  :  , :  ,  :  | ||||
| 30 | instantiation | 113, 114, 50 | ⊢  | |
: , :  , : | ||||
| 31 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_disassociation | ||||
| 32 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.posnat1 | ||||
| 33 | instantiation | 122 | ⊢  | |
| : , : | ||||
| 34 | instantiation | 101, 51, 52 | , , , , , ⊢  | |
: , : , :  | ||||
| 35 | theorem | ⊢ | ||
| proveit.logic.equality.four_chain_transitivity | ||||
| 36 | instantiation | 54, 53, 69, 120, 76, 72, 87 | , , , , , ⊢  | |
: , : , : , :  , : | ||||
| 37 | instantiation | 54, 71, 70, 124, 58, 55 | , , , , , ⊢  | |
: , : , : , :  , : | ||||
| 38 | instantiation | 56, 114, 70, 57, 124, 120, 58, 59 | , , , , , ⊢  | |
: , : , :  , : | ||||
| 39 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.scalar_mult_factorization | ||||
| 40 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.nat4 | ||||
| 41 | instantiation | 60 | ⊢  | |
| : , : , : , : | ||||
| 42 | instantiation | 101, 61, 62 | , , , , , ⊢  | |
: , : , :  | ||||
| 43 | instantiation | 101, 63, 64 | , , , , , ⊢  | |
: , : , :  | ||||
| 44 | instantiation | 106, 125, 126, 65, 108 | , , , , , ⊢  | |
: , : , :  , : | ||||
| 45 | instantiation | 113, 82, 66 | ⊢  | |
: , :  , : | ||||
| 46 | theorem | ⊢  | ||
| proveit.logic.equality.sub_right_side_into | ||||
| 47 | instantiation | 95, 67, 120 | , , ⊢  | |
:  , : | ||||
| 48 | instantiation | 68, 69, 70, 71, 72, 111, 124, 118, 120 | , , ⊢  | |
 : , : ,  : , : , : , : | ||||
| 49 | instantiation | 116, 125, 126, 73 | , , , , ⊢  | |
| : , : , : | ||||
| 50 | instantiation | 122 | ⊢  | |
| : , : | ||||
| 51 | instantiation | 74, 120, 75 | , , , , , ⊢  | |
: , :  | ||||
| 52 | instantiation | 113, 82, 76 | ⊢  | |
| : , : , : | ||||
| 53 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.nat3 | ||||
| 54 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_pulling_scalar_out_front | ||||
| 55 | instantiation | 101, 77, 78 | , , , , ⊢  | |
: , :  , :  | ||||
| 56 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_scalar_association | ||||
| 57 | instantiation | 122 | ⊢  | |
| : , : | ||||
| 58 | instantiation | 122 | ⊢  | |
| : , : | ||||
| 59 | instantiation | 106, 125, 126, 127, 108 | , , , ⊢  | |
| : , : , : , : | ||||
| 60 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.tuple_len_4_typical_eq | ||||
| 61 | instantiation | 106, 125, 126, 79, 108 | , , , , , ⊢  | |
: , : , :  , : | ||||
| 62 | instantiation | 113, 82, 80 | ⊢  | |
: , :  , : | ||||
| 63 | instantiation | 106, 125, 126, 81, 108 | , , , , , ⊢  | |
: , : , :  , : | ||||
| 64 | instantiation | 113, 82, 83 | ⊢  | |
: , :  , : | ||||
| 65 | instantiation | 116, 125, 126, 84 | , , , , ⊢  | |
| : , : , : | ||||
| 66 | instantiation | 92 | ⊢  | |
| : , : , : | ||||
| 67 | instantiation | 95, 124, 118 | , ⊢  | |
| : , : | ||||
| 68 | theorem | ⊢  | ||
| proveit.numbers.multiplication.disassociation | ||||
| 69 | axiom | ⊢  | ||
| proveit.numbers.number_sets.natural_numbers.zero_in_nats | ||||
| 70 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.nat2 | ||||
| 71 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.nat1 | ||||
| 72 | theorem | ⊢  | ||
| proveit.core_expr_types.tuples.tuple_len_0_typical_eq | ||||
| 73 | instantiation | 123, 85, 125, 126, 127 | , , , , ⊢  | |
| : , : , : , : | ||||
| 74 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_complex_right_closure | ||||
| 75 | instantiation | 86, 87 | , , , , ⊢  | |
| : | ||||
| 76 | instantiation | 92 | ⊢  | |
| : , : , : | ||||
| 77 | instantiation | 106, 125, 126, 88, 108 | , , , , ⊢  | |
: , : , :  , : | ||||
| 78 | instantiation | 113, 114, 89 | ⊢  | |
: , :  , : | ||||
| 79 | instantiation | 116, 125, 126, 90 | , , , , ⊢  | |
| : , : , : | ||||
| 80 | instantiation | 92 | ⊢  | |
| : , : , : | ||||
| 81 | instantiation | 116, 125, 126, 91 | , , , , ⊢  | |
| : , : , : | ||||
| 82 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.posnat3 | ||||
| 83 | instantiation | 92 | ⊢  | |
| : , : , : | ||||
| 84 | instantiation | 101, 93, 94 | , , , , ⊢  | |
:  , : , :  | ||||
| 85 | instantiation | 95, 124, 120 | , ⊢  | |
| : , : | ||||
| 86 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_nested_closure | ||||
| 87 | instantiation | 101, 96, 97 | , , , , ⊢  | |
: , : , :  | ||||
| 88 | instantiation | 116, 125, 126, 98 | , , , ⊢  | |
| : , : , : | ||||
| 89 | instantiation | 122 | ⊢  | |
| : , : | ||||
| 90 | instantiation | 101, 99, 100 | , , , , ⊢  | |
: , : , :  | ||||
| 91 | instantiation | 101, 102, 103 | , , , , ⊢  | |
: , : , :  | ||||
| 92 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.tuple_len_3_typical_eq | ||||
| 93 | instantiation | 123, 104, 125, 126, 127 | , , , , ⊢  | |
:  , : , : , : | ||||
| 94 | instantiation | 113, 114, 105 | ⊢  | |
| : , : , : | ||||
| 95 | theorem | ⊢  | ||
| proveit.numbers.multiplication.mult_complex_closure_bin | ||||
| 96 | instantiation | 106, 125, 126, 107, 108 | , , , , ⊢  | |
: , : , :  , : | ||||
| 97 | instantiation | 113, 114, 109 | ⊢  | |
: , :  , : | ||||
| 98 | instantiation | 123, 120, 125, 126, 127 | , , , ⊢  | |
| : , : , : , : | ||||
| 99 | instantiation | 123, 110, 125, 126, 127 | , , , , ⊢  | |
:  , : , : , : | ||||
| 100 | instantiation | 113, 114, 111 | ⊢  | |
| : , : , : | ||||
| 101 | theorem | ⊢  | ||
| proveit.logic.equality.sub_left_side_into | ||||
| 102 | instantiation | 123, 112, 125, 126, 127 | , , , , ⊢  | |
:  , : , : , : | ||||
| 103 | instantiation | 113, 114, 115 | ⊢  | |
| : , : , : | ||||
| 104 | instantiation | 119, 118, 121, 120 | , ⊢  | |
: , :  , : | ||||
| 105 | instantiation | 122 | ⊢  | |
| : , : | ||||
| 106 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_op_ket_in_QmultCodomain | ||||
| 107 | instantiation | 116, 125, 126, 117 | , , , ⊢  | |
| : , : , : | ||||
| 108 | assumption | ⊢ | ||
| 109 | instantiation | 122 | ⊢  | |
| : , : | ||||
| 110 | instantiation | 119, 124, 121, 118 | , ⊢  | |
| : , : , : | ||||
| 111 | instantiation | 122 | ⊢  | |
| : , : | ||||
| 112 | instantiation | 119, 120, 121, 124 | , ⊢  | |
| : , : , : | ||||
| 113 | axiom | ⊢  | ||
| proveit.physics.quantum.algebra.multi_qmult_def | ||||
| 114 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.posnat2 | ||||
| 115 | instantiation | 122 | ⊢  | |
| : , : | ||||
| 116 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_op_is_linmap | ||||
| 117 | instantiation | 123, 124, 125, 126, 127 | , , , ⊢  | |
| : , : , : , : | ||||
| 118 | assumption | ⊢ | ||
| 119 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_complex_ket_closure | ||||
| 120 | assumption | ⊢ | ||
| 121 | theorem | ⊢  | ||
| proveit.linear_algebra.inner_products.complex_set_is_hilbert_space | ||||
| 122 | theorem | ⊢  | ||
| proveit.numbers.numerals.decimals.tuple_len_2_typical_eq | ||||
| 123 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_complex_op_closure | ||||
| 124 | assumption | ⊢ | ||
| 125 | instantiation | 128, 131 | ⊢  | |
| : | ||||
| 126 | instantiation | 128, 130 | ⊢  | |
| : | ||||
| 127 | instantiation | 129, 130, 131, 132 | , , ⊢  | |
: , : ,  : | ||||
| 128 | theorem | ⊢  | ||
| proveit.linear_algebra.inner_products.complex_vec_set_is_hilbert_space | ||||
| 129 | theorem | ⊢  | ||
| proveit.physics.quantum.algebra.qmult_matrix_is_linmap | ||||
| 130 | assumption | ⊢ | ||
| 131 | assumption | ⊢ | ||
| 132 | assumption | ⊢ | ||
| *equality replacement requirements | ||||