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 Conditional, ExprRange, ExprTuple, IndexedVar, Lambda, Variable, i, lambda_, m, n
from proveit.core_expr_types import a_i, b_1_to_n, b_i, lambda_i
from proveit.linear_algebra import OrthoNormBases, ScalarMult, VecSum
from proveit.logic import CartExp, Equals, Exists, InSet, Set
from proveit.numbers import Complex, Interval, Min, RealNonNeg, one
from proveit.physics.quantum import Ket, Qmult, var_ket_psi
from proveit.physics.quantum.algebra import b_1_to_n_kets

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Min(m, n)
expr = ExprTuple(Lambda([b_1_to_n], Conditional(Exists(instance_param_or_params = [ExprRange(sub_expr1, IndexedVar(lambda_, sub_expr1), one, sub_expr2)], instance_expr = Equals(var_ket_psi, VecSum(index_or_indices = [i], summand = ScalarMult(lambda_i, Qmult(Ket(a_i), Ket(b_i))), domain = Interval(one, sub_expr2))), domain = RealNonNeg).with_wrapping(), InSet(Set(b_1_to_n_kets), OrthoNormBases(CartExp(Complex, n))))))

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(b_{1}, b_{2}, \ldots, b_{n}\right) \mapsto \left\{\begin{array}{l}\exists_{\lambda_{1}, \lambda_{2}, \ldots, \lambda_{{\rm Min}\left(m, n\right)} \in \mathbb{R}^{\ge 0}}~\\
\left(\lvert \psi \rangle = \left(\sum_{i=1}^{{\rm Min}\left(m, n\right)} \left(\lambda_{i} \cdot \left(\lvert a_{i} \rangle \thinspace \lvert b_{i} \rangle\right)\right)\right)\right)\end{array} \textrm{ if } \left\{\left\{\lvert b_{1} \rangle, \lvert b_{2} \rangle, \ldots, \lvert b_{n} \rangle\right\}\right\} \in \textrm{O.N.Bases}\left(\mathbb{C}^{n}\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	parameters: 2 body: 3
2	ExprTuple	4
3	Conditional	value: 5 condition: 6
4	ExprRange	lambda_map: 7 start_index: 69 end_index: 79
5	Operation	operator: 8 operand: 11
6	Operation	operator: 55 operands: 10
7	Lambda	parameter: 61 body: 52
8	Literal
9	ExprTuple	11
10	ExprTuple	12, 13
11	Lambda	parameters: 14 body: 15
12	Operation	operator: 29 operand: 22
13	Operation	operator: 17 operand: 23
14	ExprTuple	19
15	Conditional	value: 20 condition: 21
16	ExprTuple	22
17	Literal
18	ExprTuple	23
19	ExprRange	lambda_map: 24 start_index: 69 end_index: 70
20	Operation	operator: 25 operands: 26
21	Operation	operator: 27 operands: 28
22	Operation	operator: 29 operands: 30
23	Operation	operator: 31 operands: 32
24	Lambda	parameter: 61 body: 50
25	Literal
26	ExprTuple	33, 34
27	Literal
28	ExprTuple	35
29	Literal
30	ExprTuple	36
31	Literal
32	ExprTuple	37, 79
33	Variable
34	Operation	operator: 38 operand: 42
35	ExprRange	lambda_map: 40 start_index: 69 end_index: 70
36	ExprRange	lambda_map: 41 start_index: 69 end_index: 79
37	Literal
38	Literal
39	ExprTuple	42
40	Lambda	parameter: 61 body: 43
41	Lambda	parameter: 61 body: 44
42	Lambda	parameter: 83 body: 45
43	Operation	operator: 55 operands: 46
44	Operation	operator: 72 operand: 52
45	Conditional	value: 48 condition: 49
46	ExprTuple	50, 51
47	ExprTuple	52
48	Operation	operator: 53 operands: 54
49	Operation	operator: 55 operands: 56
50	IndexedVar	variable: 62 index: 61
51	Literal
52	IndexedVar	variable: 81 index: 61
53	Literal
54	ExprTuple	58, 59
55	Literal
56	ExprTuple	83, 60
57	ExprTuple	61
58	IndexedVar	variable: 62 index: 83
59	Operation	operator: 63 operands: 64
60	Operation	operator: 65 operands: 66
61	Variable
62	Variable
63	Literal
64	ExprTuple	67, 68
65	Literal
66	ExprTuple	69, 70
67	Operation	operator: 72 operand: 76
68	Operation	operator: 72 operand: 77
69	Literal
70	Operation	operator: 74 operands: 75
71	ExprTuple	76
72	Literal
73	ExprTuple	77
74	Literal
75	ExprTuple	78, 79
76	IndexedVar	variable: 80 index: 83
77	IndexedVar	variable: 81 index: 83
78	Variable
79	Variable
80	Variable
81	Variable
82	ExprTuple	83
83	Variable