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 OrthoNormBases, VecSum
from proveit.logic import And, Equals, Forall, 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 = [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(), And(InSet(m, NaturalPos), InSet(n, NaturalPos)))

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_{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} \textrm{ if } m \in \mathbb{N}^+ ,  n \in \mathbb{N}^+\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: 18 operand: 6
2	Operation	operator: 4 operands: 5
3	ExprTuple	6
4	Literal
5	ExprTuple	7, 8
6	Lambda	parameters: 9 body: 10
7	Operation	operator: 73 operands: 11
8	Operation	operator: 73 operands: 12
9	ExprTuple	13
10	Conditional	value: 14 condition: 15
11	ExprTuple	69, 16
12	ExprTuple	90, 16
13	ExprRange	lambda_map: 17 start_index: 89 end_index: 69
14	Operation	operator: 18 operand: 21
15	Operation	operator: 73 operands: 20
16	Literal
17	Lambda	parameter: 70 body: 51
18	Literal
19	ExprTuple	21
20	ExprTuple	22, 23
21	Lambda	parameters: 24 body: 25
22	Operation	operator: 43 operands: 26
23	Operation	operator: 45 operand: 32
24	ExprTuple	28
25	Conditional	value: 29 condition: 30
26	ExprTuple	31
27	ExprTuple	32
28	ExprRange	lambda_map: 33 start_index: 89 end_index: 90
29	Operation	operator: 34 operands: 35
30	Operation	operator: 73 operands: 36
31	ExprRange	lambda_map: 37 start_index: 89 end_index: 69
32	Variable
33	Lambda	parameter: 70 body: 63
34	Literal
35	ExprTuple	76, 38
36	ExprTuple	39, 40
37	Lambda	parameter: 70 body: 41
38	Operation	operator: 57 operand: 48
39	Operation	operator: 43 operands: 44
40	Operation	operator: 45 operand: 50
41	Operation	operator: 81 operand: 51
42	ExprTuple	48
43	Literal
44	ExprTuple	49
45	Literal
46	ExprTuple	50
47	ExprTuple	51
48	Lambda	parameter: 96 body: 52
49	ExprRange	lambda_map: 53 start_index: 89 end_index: 90
50	Variable
51	IndexedVar	variable: 93 index: 70
52	Conditional	value: 54 condition: 55
53	Lambda	parameter: 70 body: 56
54	Operation	operator: 57 operand: 61
55	Operation	operator: 73 operands: 59
56	Operation	operator: 81 operand: 63
57	Literal
58	ExprTuple	61
59	ExprTuple	96, 62
60	ExprTuple	63
61	Lambda	parameter: 95 body: 64
62	Operation	operator: 85 operands: 65
63	IndexedVar	variable: 91 index: 70
64	Conditional	value: 67 condition: 68
65	ExprTuple	89, 69
66	ExprTuple	70
67	Operation	operator: 71 operands: 72
68	Operation	operator: 73 operands: 74
69	Variable
70	Variable
71	Literal
72	ExprTuple	75, 76, 77, 78, 79
73	Literal
74	ExprTuple	95, 80
75	Operation	operator: 83 operand: 87
76	Variable
77	Operation	operator: 81 operand: 88
78	Operation	operator: 81 operand: 87
79	Operation	operator: 83 operand: 88
80	Operation	operator: 85 operands: 86
81	Literal
82	ExprTuple	87
83	Literal
84	ExprTuple	88
85	Literal
86	ExprTuple	89, 90
87	IndexedVar	variable: 91 index: 95
88	IndexedVar	variable: 93 index: 96
89	Literal
90	Variable
91	Variable
92	ExprTuple	95
93	Variable
94	ExprTuple	96
95	Variable
96	Variable