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