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_1_to_m, 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 a_1_to_m_kets, 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([a_1_to_m], Conditional(Exists(instance_param_or_params = [b_1_to_n], instance_expr = 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(), condition = InSet(Set(b_1_to_n_kets), OrthoNormBases(CartExp(Complex, n)))).with_wrapping(), InSet(Set(a_1_to_m_kets), OrthoNormBases(CartExp(Complex, m))))))

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(a_{1}, a_{2}, \ldots, a_{m}\right) \mapsto \left\{\begin{array}{l}\exists_{b_{1}, b_{2}, \ldots, b_{n}~|~\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)}~\\
\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}\right]\end{array} \textrm{ if } \left\{\left\{\lvert a_{1} \rangle, \lvert a_{2} \rangle, \ldots, \lvert a_{m} \rangle\right\}\right\} \in \textrm{O.N.Bases}\left(\mathbb{C}^{m}\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: 91 end_index: 100
5	Operation	operator: 23 operand: 10
6	Operation	operator: 77 operands: 9
7	Lambda	parameter: 83 body: 59
8	ExprTuple	10
9	ExprTuple	11, 12
10	Lambda	parameters: 13 body: 14
11	Operation	operator: 49 operand: 20
12	Operation	operator: 35 operand: 21
13	ExprTuple	17
14	Conditional	value: 18 condition: 19
15	ExprTuple	20
16	ExprTuple	21
17	ExprRange	lambda_map: 22 start_index: 91 end_index: 101
18	Operation	operator: 23 operand: 28
19	Operation	operator: 77 operands: 25
20	Operation	operator: 49 operands: 26
21	Operation	operator: 51 operands: 27
22	Lambda	parameter: 83 body: 74
23	Literal
24	ExprTuple	28
25	ExprTuple	29, 30
26	ExprTuple	31
27	ExprTuple	58, 100
28	Lambda	parameters: 32 body: 33
29	Operation	operator: 49 operand: 41
30	Operation	operator: 35 operand: 42
31	ExprRange	lambda_map: 37 start_index: 91 end_index: 100
32	ExprTuple	38
33	Conditional	value: 39 condition: 40
34	ExprTuple	41
35	Literal
36	ExprTuple	42
37	Lambda	parameter: 83 body: 43
38	ExprRange	lambda_map: 44 start_index: 91 end_index: 92
39	Operation	operator: 45 operands: 46
40	Operation	operator: 47 operands: 48
41	Operation	operator: 49 operands: 50
42	Operation	operator: 51 operands: 52
43	Operation	operator: 94 operand: 59
44	Lambda	parameter: 83 body: 72
45	Literal
46	ExprTuple	54, 55
47	Literal
48	ExprTuple	56
49	Literal
50	ExprTuple	57
51	Literal
52	ExprTuple	58, 101
53	ExprTuple	59
54	Variable
55	Operation	operator: 60 operand: 64
56	ExprRange	lambda_map: 62 start_index: 91 end_index: 92
57	ExprRange	lambda_map: 63 start_index: 91 end_index: 101
58	Literal
59	IndexedVar	variable: 102 index: 83
60	Literal
61	ExprTuple	64
62	Lambda	parameter: 83 body: 65
63	Lambda	parameter: 83 body: 66
64	Lambda	parameter: 105 body: 67
65	Operation	operator: 77 operands: 68
66	Operation	operator: 94 operand: 74
67	Conditional	value: 70 condition: 71
68	ExprTuple	72, 73
69	ExprTuple	74
70	Operation	operator: 75 operands: 76
71	Operation	operator: 77 operands: 78
72	IndexedVar	variable: 84 index: 83
73	Literal
74	IndexedVar	variable: 103 index: 83
75	Literal
76	ExprTuple	80, 81
77	Literal
78	ExprTuple	105, 82
79	ExprTuple	83
80	IndexedVar	variable: 84 index: 105
81	Operation	operator: 85 operands: 86
82	Operation	operator: 87 operands: 88
83	Variable
84	Variable
85	Literal
86	ExprTuple	89, 90
87	Literal
88	ExprTuple	91, 92
89	Operation	operator: 94 operand: 98
90	Operation	operator: 94 operand: 99
91	Literal
92	Operation	operator: 96 operands: 97
93	ExprTuple	98
94	Literal
95	ExprTuple	99
96	Literal
97	ExprTuple	100, 101
98	IndexedVar	variable: 102 index: 105
99	IndexedVar	variable: 103 index: 105
100	Variable
101	Variable
102	Variable
103	Variable
104	ExprTuple	105
105	Variable