Expression of type Lambda

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, Lambda, W, i, j, m, n
from proveit.core_expr_types import v_i, w_1_to_n, w_j
from proveit.linear_algebra import OrthoNormBases, VecSum
from proveit.logic import Equals, InSet
from proveit.numbers import Interval, one
from proveit.physics.quantum import Bra, Ket, Qmult
from proveit.physics.quantum.algebra import w_1_to_n_kets

# build up the expression from sub-expressions
expr = Lambda([w_1_to_n], Conditional(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))), InSet(w_1_to_n_kets, OrthoNormBases(W))))

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(w_{1}, w_{2}, \ldots, w_{n}\right) \mapsto \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) \textrm{ if } \left\{\lvert w_{1} \rangle, \lvert w_{2} \rangle, \ldots, \lvert w_{n} \rangle\right\} \in \textrm{O.N.Bases}\left(W\right)\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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