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 HilbertSpaces, LinMap, OrthoNormBases, VecSum
from proveit.logic import And, Equals, Forall, InClass, 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([V, W], Conditional(Forall(instance_param_or_params = [A], instance_expr = 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), domain = LinMap(V, W)), And(InClass(V, HilbertSpaces), InClass(W, HilbertSpaces)))))

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(V, W\right) \mapsto \left\{\forall_{A \in \mathcal{L}\left(V, W\right)}~\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]\right] \textrm{ if } V \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces} ,  W \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces}\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: 25 body: 2
2	Conditional	value: 3 condition: 4
3	Operation	operator: 43 operand: 7
4	Operation	operator: 29 operands: 6
5	ExprTuple	7
6	ExprTuple	8, 9
7	Lambda	parameter: 101 body: 11
8	Operation	operator: 13 operands: 12
9	Operation	operator: 13 operands: 14
10	ExprTuple	101
11	Conditional	value: 15 condition: 16
12	ExprTuple	57, 17
13	Literal
14	ExprTuple	75, 17
15	Operation	operator: 43 operand: 20
16	Operation	operator: 98 operands: 19
17	Literal
18	ExprTuple	20
19	ExprTuple	101, 21
20	Lambda	parameters: 22 body: 23
21	Operation	operator: 24 operands: 25
22	ExprTuple	94, 115
23	Conditional	value: 26 condition: 27
24	Literal
25	ExprTuple	57, 75
26	Operation	operator: 43 operand: 31
27	Operation	operator: 29 operands: 30
28	ExprTuple	31
29	Literal
30	ExprTuple	32, 33
31	Lambda	parameters: 34 body: 35
32	Operation	operator: 98 operands: 36
33	Operation	operator: 98 operands: 37
34	ExprTuple	38
35	Conditional	value: 39 condition: 40
36	ExprTuple	94, 41
37	ExprTuple	115, 41
38	ExprRange	lambda_map: 42 start_index: 114 end_index: 94
39	Operation	operator: 43 operand: 46
40	Operation	operator: 98 operands: 45
41	Literal
42	Lambda	parameter: 95 body: 76
43	Literal
44	ExprTuple	46
45	ExprTuple	47, 48
46	Lambda	parameters: 49 body: 50
47	Operation	operator: 68 operands: 51
48	Operation	operator: 70 operand: 57
49	ExprTuple	53
50	Conditional	value: 54 condition: 55
51	ExprTuple	56
52	ExprTuple	57
53	ExprRange	lambda_map: 58 start_index: 114 end_index: 115
54	Operation	operator: 59 operands: 60
55	Operation	operator: 98 operands: 61
56	ExprRange	lambda_map: 62 start_index: 114 end_index: 94
57	Variable
58	Lambda	parameter: 95 body: 88
59	Literal
60	ExprTuple	101, 63
61	ExprTuple	64, 65
62	Lambda	parameter: 95 body: 66
63	Operation	operator: 82 operand: 73
64	Operation	operator: 68 operands: 69
65	Operation	operator: 70 operand: 75
66	Operation	operator: 106 operand: 76
67	ExprTuple	73
68	Literal
69	ExprTuple	74
70	Literal
71	ExprTuple	75
72	ExprTuple	76
73	Lambda	parameter: 121 body: 77
74	ExprRange	lambda_map: 78 start_index: 114 end_index: 115
75	Variable
76	IndexedVar	variable: 118 index: 95
77	Conditional	value: 79 condition: 80
78	Lambda	parameter: 95 body: 81
79	Operation	operator: 82 operand: 86
80	Operation	operator: 98 operands: 84
81	Operation	operator: 106 operand: 88
82	Literal
83	ExprTuple	86
84	ExprTuple	121, 87
85	ExprTuple	88
86	Lambda	parameter: 120 body: 89
87	Operation	operator: 110 operands: 90
88	IndexedVar	variable: 116 index: 95
89	Conditional	value: 92 condition: 93
90	ExprTuple	114, 94
91	ExprTuple	95
92	Operation	operator: 96 operands: 97
93	Operation	operator: 98 operands: 99
94	Variable
95	Variable
96	Literal
97	ExprTuple	100, 101, 102, 103, 104
98	Literal
99	ExprTuple	120, 105
100	Operation	operator: 108 operand: 112
101	Variable
102	Operation	operator: 106 operand: 113
103	Operation	operator: 106 operand: 112
104	Operation	operator: 108 operand: 113
105	Operation	operator: 110 operands: 111
106	Literal
107	ExprTuple	112
108	Literal
109	ExprTuple	113
110	Literal
111	ExprTuple	114, 115
112	IndexedVar	variable: 116 index: 120
113	IndexedVar	variable: 118 index: 121
114	Literal
115	Variable
116	Variable
117	ExprTuple	120
118	Variable
119	ExprTuple	121
120	Variable
121	Variable