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 LinMap, OrthoNormBases, VecSum
from proveit.logic import 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 = [m, n], instance_expr = 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(), domain = NaturalPos), InSet(A, LinMap(V, 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\{\forall_{m, n \in \mathbb{N}^+}~\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}\right] \textrm{ if } A \in \mathcal{L}\left(V, W\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: 28 operand: 5
2	Operation	operator: 83 operands: 4
3	ExprTuple	5
4	ExprTuple	86, 6
5	Lambda	parameters: 7 body: 8
6	Operation	operator: 9 operands: 10
7	ExprTuple	79, 100
8	Conditional	value: 11 condition: 12
9	Literal
10	ExprTuple	42, 60
11	Operation	operator: 28 operand: 16
12	Operation	operator: 14 operands: 15
13	ExprTuple	16
14	Literal
15	ExprTuple	17, 18
16	Lambda	parameters: 19 body: 20
17	Operation	operator: 83 operands: 21
18	Operation	operator: 83 operands: 22
19	ExprTuple	23
20	Conditional	value: 24 condition: 25
21	ExprTuple	79, 26
22	ExprTuple	100, 26
23	ExprRange	lambda_map: 27 start_index: 99 end_index: 79
24	Operation	operator: 28 operand: 31
25	Operation	operator: 83 operands: 30
26	Literal
27	Lambda	parameter: 80 body: 61
28	Literal
29	ExprTuple	31
30	ExprTuple	32, 33
31	Lambda	parameters: 34 body: 35
32	Operation	operator: 53 operands: 36
33	Operation	operator: 55 operand: 42
34	ExprTuple	38
35	Conditional	value: 39 condition: 40
36	ExprTuple	41
37	ExprTuple	42
38	ExprRange	lambda_map: 43 start_index: 99 end_index: 100
39	Operation	operator: 44 operands: 45
40	Operation	operator: 83 operands: 46
41	ExprRange	lambda_map: 47 start_index: 99 end_index: 79
42	Variable
43	Lambda	parameter: 80 body: 73
44	Literal
45	ExprTuple	86, 48
46	ExprTuple	49, 50
47	Lambda	parameter: 80 body: 51
48	Operation	operator: 67 operand: 58
49	Operation	operator: 53 operands: 54
50	Operation	operator: 55 operand: 60
51	Operation	operator: 91 operand: 61
52	ExprTuple	58
53	Literal
54	ExprTuple	59
55	Literal
56	ExprTuple	60
57	ExprTuple	61
58	Lambda	parameter: 106 body: 62
59	ExprRange	lambda_map: 63 start_index: 99 end_index: 100
60	Variable
61	IndexedVar	variable: 103 index: 80
62	Conditional	value: 64 condition: 65
63	Lambda	parameter: 80 body: 66
64	Operation	operator: 67 operand: 71
65	Operation	operator: 83 operands: 69
66	Operation	operator: 91 operand: 73
67	Literal
68	ExprTuple	71
69	ExprTuple	106, 72
70	ExprTuple	73
71	Lambda	parameter: 105 body: 74
72	Operation	operator: 95 operands: 75
73	IndexedVar	variable: 101 index: 80
74	Conditional	value: 77 condition: 78
75	ExprTuple	99, 79
76	ExprTuple	80
77	Operation	operator: 81 operands: 82
78	Operation	operator: 83 operands: 84
79	Variable
80	Variable
81	Literal
82	ExprTuple	85, 86, 87, 88, 89
83	Literal
84	ExprTuple	105, 90
85	Operation	operator: 93 operand: 97
86	Variable
87	Operation	operator: 91 operand: 98
88	Operation	operator: 91 operand: 97
89	Operation	operator: 93 operand: 98
90	Operation	operator: 95 operands: 96
91	Literal
92	ExprTuple	97
93	Literal
94	ExprTuple	98
95	Literal
96	ExprTuple	99, 100
97	IndexedVar	variable: 101 index: 105
98	IndexedVar	variable: 103 index: 106
99	Literal
100	Variable
101	Variable
102	ExprTuple	105
103	Variable
104	ExprTuple	106
105	Variable
106	Variable