Expression of type ExprTuple

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, ExprTuple, Lambda, 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 OrthoNormBases, VecSum
from proveit.logic import Equals, Forall, InSet
from proveit.numbers import Interval, 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 = ExprTuple(Lambda([v_1_to_m], Conditional(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(), InSet(v_1_to_m_kets, OrthoNormBases(V)))))

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(\left(v_{1}, v_{2}, \ldots, v_{m}\right) \mapsto \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} \textrm{ if } \left\{\lvert v_{1} \rangle, \lvert v_{2} \rangle, \ldots, \lvert v_{m} \rangle\right\} \in \textrm{O.N.Bases}\left(V\right)\right..\right)

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	ExprTuple	1
1	Lambda	parameters: 2 body: 3
2	ExprTuple	4
3	Conditional	value: 5 condition: 6
4	ExprRange	lambda_map: 7 start_index: 79 end_index: 59
5	Operation	operator: 8 operand: 11
6	Operation	operator: 63 operands: 10
7	Lambda	parameter: 60 body: 41
8	Literal
9	ExprTuple	11
10	ExprTuple	12, 13
11	Lambda	parameters: 14 body: 15
12	Operation	operator: 33 operands: 16
13	Operation	operator: 35 operand: 22
14	ExprTuple	18
15	Conditional	value: 19 condition: 20
16	ExprTuple	21
17	ExprTuple	22
18	ExprRange	lambda_map: 23 start_index: 79 end_index: 80
19	Operation	operator: 24 operands: 25
20	Operation	operator: 63 operands: 26
21	ExprRange	lambda_map: 27 start_index: 79 end_index: 59
22	Variable
23	Lambda	parameter: 60 body: 53
24	Literal
25	ExprTuple	66, 28
26	ExprTuple	29, 30
27	Lambda	parameter: 60 body: 31
28	Operation	operator: 47 operand: 38
29	Operation	operator: 33 operands: 34
30	Operation	operator: 35 operand: 40
31	Operation	operator: 71 operand: 41
32	ExprTuple	38
33	Literal
34	ExprTuple	39
35	Literal
36	ExprTuple	40
37	ExprTuple	41
38	Lambda	parameter: 86 body: 42
39	ExprRange	lambda_map: 43 start_index: 79 end_index: 80
40	Variable
41	IndexedVar	variable: 83 index: 60
42	Conditional	value: 44 condition: 45
43	Lambda	parameter: 60 body: 46
44	Operation	operator: 47 operand: 51
45	Operation	operator: 63 operands: 49
46	Operation	operator: 71 operand: 53
47	Literal
48	ExprTuple	51
49	ExprTuple	86, 52
50	ExprTuple	53
51	Lambda	parameter: 85 body: 54
52	Operation	operator: 75 operands: 55
53	IndexedVar	variable: 81 index: 60
54	Conditional	value: 57 condition: 58
55	ExprTuple	79, 59
56	ExprTuple	60
57	Operation	operator: 61 operands: 62
58	Operation	operator: 63 operands: 64
59	Variable
60	Variable
61	Literal
62	ExprTuple	65, 66, 67, 68, 69
63	Literal
64	ExprTuple	85, 70
65	Operation	operator: 73 operand: 77
66	Variable
67	Operation	operator: 71 operand: 78
68	Operation	operator: 71 operand: 77
69	Operation	operator: 73 operand: 78
70	Operation	operator: 75 operands: 76
71	Literal
72	ExprTuple	77
73	Literal
74	ExprTuple	78
75	Literal
76	ExprTuple	79, 80
77	IndexedVar	variable: 81 index: 85
78	IndexedVar	variable: 83 index: 86
79	Literal
80	Variable
81	Variable
82	ExprTuple	85
83	Variable
84	ExprTuple	86
85	Variable
86	Variable