Expression of type ExprTuple

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, ExprTuple, 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 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 = ExprTuple(Lambda([m, n], 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(\left(m, n\right) \mapsto \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..\right)

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	ExprTuple	1
1	Lambda	parameters: 2 body: 3
2	ExprTuple	72, 93
3	Conditional	value: 4 condition: 5
4	Operation	operator: 21 operand: 9
5	Operation	operator: 7 operands: 8
6	ExprTuple	9
7	Literal
8	ExprTuple	10, 11
9	Lambda	parameters: 12 body: 13
10	Operation	operator: 76 operands: 14
11	Operation	operator: 76 operands: 15
12	ExprTuple	16
13	Conditional	value: 17 condition: 18
14	ExprTuple	72, 19
15	ExprTuple	93, 19
16	ExprRange	lambda_map: 20 start_index: 92 end_index: 72
17	Operation	operator: 21 operand: 24
18	Operation	operator: 76 operands: 23
19	Literal
20	Lambda	parameter: 73 body: 54
21	Literal
22	ExprTuple	24
23	ExprTuple	25, 26
24	Lambda	parameters: 27 body: 28
25	Operation	operator: 46 operands: 29
26	Operation	operator: 48 operand: 35
27	ExprTuple	31
28	Conditional	value: 32 condition: 33
29	ExprTuple	34
30	ExprTuple	35
31	ExprRange	lambda_map: 36 start_index: 92 end_index: 93
32	Operation	operator: 37 operands: 38
33	Operation	operator: 76 operands: 39
34	ExprRange	lambda_map: 40 start_index: 92 end_index: 72
35	Variable
36	Lambda	parameter: 73 body: 66
37	Literal
38	ExprTuple	79, 41
39	ExprTuple	42, 43
40	Lambda	parameter: 73 body: 44
41	Operation	operator: 60 operand: 51
42	Operation	operator: 46 operands: 47
43	Operation	operator: 48 operand: 53
44	Operation	operator: 84 operand: 54
45	ExprTuple	51
46	Literal
47	ExprTuple	52
48	Literal
49	ExprTuple	53
50	ExprTuple	54
51	Lambda	parameter: 99 body: 55
52	ExprRange	lambda_map: 56 start_index: 92 end_index: 93
53	Variable
54	IndexedVar	variable: 96 index: 73
55	Conditional	value: 57 condition: 58
56	Lambda	parameter: 73 body: 59
57	Operation	operator: 60 operand: 64
58	Operation	operator: 76 operands: 62
59	Operation	operator: 84 operand: 66
60	Literal
61	ExprTuple	64
62	ExprTuple	99, 65
63	ExprTuple	66
64	Lambda	parameter: 98 body: 67
65	Operation	operator: 88 operands: 68
66	IndexedVar	variable: 94 index: 73
67	Conditional	value: 70 condition: 71
68	ExprTuple	92, 72
69	ExprTuple	73
70	Operation	operator: 74 operands: 75
71	Operation	operator: 76 operands: 77
72	Variable
73	Variable
74	Literal
75	ExprTuple	78, 79, 80, 81, 82
76	Literal
77	ExprTuple	98, 83
78	Operation	operator: 86 operand: 90
79	Variable
80	Operation	operator: 84 operand: 91
81	Operation	operator: 84 operand: 90
82	Operation	operator: 86 operand: 91
83	Operation	operator: 88 operands: 89
84	Literal
85	ExprTuple	90
86	Literal
87	ExprTuple	91
88	Literal
89	ExprTuple	92, 93
90	IndexedVar	variable: 94 index: 98
91	IndexedVar	variable: 96 index: 99
92	Literal
93	Variable
94	Variable
95	ExprTuple	98
96	Variable
97	ExprTuple	99
98	Variable
99	Variable