Expression of type Conditional

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, IndexedVar, Variable, i, lambda_, m, n
from proveit.core_expr_types import a_i, 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 = 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\{\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..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

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