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, 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 = 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(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..

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