Expression of type Conditional

from the theory of proveit.physics.quantum.algebra

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import A, Conditional, V, W, i, j, m, n
from proveit.core_expr_types import v_1_to_m, v_i, w_1_to_n, w_j
from proveit.linear_algebra import HilbertSpaces, LinMap, OrthoNormBases, VecSum
from proveit.logic import And, Equals, Forall, InClass, InSet
from proveit.numbers import Interval, NaturalPos, one
from proveit.physics.quantum import Bra, Ket, Qmult
from proveit.physics.quantum.algebra import v_1_to_m_kets, w_1_to_n_kets

# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [A], instance_expr = Forall(instance_param_or_params = [m, n], instance_expr = Forall(instance_param_or_params = [v_1_to_m], instance_expr = Forall(instance_param_or_params = [w_1_to_n], instance_expr = Equals(A, VecSum(index_or_indices = [i], summand = VecSum(index_or_indices = [j], summand = Qmult(Bra(w_j), A, Ket(v_i), Ket(w_j), Bra(v_i)), domain = Interval(one, n)), domain = Interval(one, m))), condition = InSet(w_1_to_n_kets, OrthoNormBases(W))).with_wrapping(), condition = InSet(v_1_to_m_kets, OrthoNormBases(V))).with_wrapping(), domain = NaturalPos), domain = LinMap(V, W)), And(InClass(V, HilbertSpaces), InClass(W, HilbertSpaces)))

expr:

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left\{\forall_{A \in \mathcal{L}\left(V, W\right)}~\left[\forall_{m, n \in \mathbb{N}^+}~\left[\begin{array}{l}\forall_{v_{1}, v_{2}, \ldots, v_{m}~|~\left\{\lvert v_{1} \rangle, \lvert v_{2} \rangle, \ldots, \lvert v_{m} \rangle\right\} \in \textrm{O.N.Bases}\left(V\right)}~\\
\left[\begin{array}{l}\forall_{w_{1}, w_{2}, \ldots, w_{n}~|~\left\{\lvert w_{1} \rangle, \lvert w_{2} \rangle, \ldots, \lvert w_{n} \rangle\right\} \in \textrm{O.N.Bases}\left(W\right)}~\\
\left(A = \left(\sum_{i=1}^{m} \left(\sum_{j=1}^{n} \left(\langle w_{j} \rvert \thinspace A \thinspace \lvert v_{i} \rangle \thinspace \lvert w_{j} \rangle \thinspace \langle v_{i} \rvert\right)\right)\right)\right)\end{array}\right]\end{array}\right]\right] \textrm{ if } V \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces} ,  W \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces}\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 41 operand: 5
2	Operation	operator: 27 operands: 4
3	ExprTuple	5
4	ExprTuple	6, 7
5	Lambda	parameter: 99 body: 9
6	Operation	operator: 11 operands: 10
7	Operation	operator: 11 operands: 12
8	ExprTuple	99
9	Conditional	value: 13 condition: 14
10	ExprTuple	55, 15
11	Literal
12	ExprTuple	73, 15
13	Operation	operator: 41 operand: 18
14	Operation	operator: 96 operands: 17
15	Literal
16	ExprTuple	18
17	ExprTuple	99, 19
18	Lambda	parameters: 20 body: 21
19	Operation	operator: 22 operands: 23
20	ExprTuple	92, 113
21	Conditional	value: 24 condition: 25
22	Literal
23	ExprTuple	55, 73
24	Operation	operator: 41 operand: 29
25	Operation	operator: 27 operands: 28
26	ExprTuple	29
27	Literal
28	ExprTuple	30, 31
29	Lambda	parameters: 32 body: 33
30	Operation	operator: 96 operands: 34
31	Operation	operator: 96 operands: 35
32	ExprTuple	36
33	Conditional	value: 37 condition: 38
34	ExprTuple	92, 39
35	ExprTuple	113, 39
36	ExprRange	lambda_map: 40 start_index: 112 end_index: 92
37	Operation	operator: 41 operand: 44
38	Operation	operator: 96 operands: 43
39	Literal
40	Lambda	parameter: 93 body: 74
41	Literal
42	ExprTuple	44
43	ExprTuple	45, 46
44	Lambda	parameters: 47 body: 48
45	Operation	operator: 66 operands: 49
46	Operation	operator: 68 operand: 55
47	ExprTuple	51
48	Conditional	value: 52 condition: 53
49	ExprTuple	54
50	ExprTuple	55
51	ExprRange	lambda_map: 56 start_index: 112 end_index: 113
52	Operation	operator: 57 operands: 58
53	Operation	operator: 96 operands: 59
54	ExprRange	lambda_map: 60 start_index: 112 end_index: 92
55	Variable
56	Lambda	parameter: 93 body: 86
57	Literal
58	ExprTuple	99, 61
59	ExprTuple	62, 63
60	Lambda	parameter: 93 body: 64
61	Operation	operator: 80 operand: 71
62	Operation	operator: 66 operands: 67
63	Operation	operator: 68 operand: 73
64	Operation	operator: 104 operand: 74
65	ExprTuple	71
66	Literal
67	ExprTuple	72
68	Literal
69	ExprTuple	73
70	ExprTuple	74
71	Lambda	parameter: 119 body: 75
72	ExprRange	lambda_map: 76 start_index: 112 end_index: 113
73	Variable
74	IndexedVar	variable: 116 index: 93
75	Conditional	value: 77 condition: 78
76	Lambda	parameter: 93 body: 79
77	Operation	operator: 80 operand: 84
78	Operation	operator: 96 operands: 82
79	Operation	operator: 104 operand: 86
80	Literal
81	ExprTuple	84
82	ExprTuple	119, 85
83	ExprTuple	86
84	Lambda	parameter: 118 body: 87
85	Operation	operator: 108 operands: 88
86	IndexedVar	variable: 114 index: 93
87	Conditional	value: 90 condition: 91
88	ExprTuple	112, 92
89	ExprTuple	93
90	Operation	operator: 94 operands: 95
91	Operation	operator: 96 operands: 97
92	Variable
93	Variable
94	Literal
95	ExprTuple	98, 99, 100, 101, 102
96	Literal
97	ExprTuple	118, 103
98	Operation	operator: 106 operand: 110
99	Variable
100	Operation	operator: 104 operand: 111
101	Operation	operator: 104 operand: 110
102	Operation	operator: 106 operand: 111
103	Operation	operator: 108 operands: 109
104	Literal
105	ExprTuple	110
106	Literal
107	ExprTuple	111
108	Literal
109	ExprTuple	112, 113
110	IndexedVar	variable: 114 index: 118
111	IndexedVar	variable: 116 index: 119
112	Literal
113	Variable
114	Variable
115	ExprTuple	118
116	Variable
117	ExprTuple	119
118	Variable
119	Variable