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_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 = 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\{\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()

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