Expression of type Lambda

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, 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 = 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(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..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameters: 1 body: 2
1	ExprTuple	3
2	Conditional	value: 4 condition: 5
3	ExprRange	lambda_map: 6 start_index: 68 end_index: 78
4	Operation	operator: 7 operand: 10
5	Operation	operator: 54 operands: 9
6	Lambda	parameter: 60 body: 51
7	Literal
8	ExprTuple	10
9	ExprTuple	11, 12
10	Lambda	parameters: 13 body: 14
11	Operation	operator: 28 operand: 21
12	Operation	operator: 16 operand: 22
13	ExprTuple	18
14	Conditional	value: 19 condition: 20
15	ExprTuple	21
16	Literal
17	ExprTuple	22
18	ExprRange	lambda_map: 23 start_index: 68 end_index: 69
19	Operation	operator: 24 operands: 25
20	Operation	operator: 26 operands: 27
21	Operation	operator: 28 operands: 29
22	Operation	operator: 30 operands: 31
23	Lambda	parameter: 60 body: 49
24	Literal
25	ExprTuple	32, 33
26	Literal
27	ExprTuple	34
28	Literal
29	ExprTuple	35
30	Literal
31	ExprTuple	36, 78
32	Variable
33	Operation	operator: 37 operand: 41
34	ExprRange	lambda_map: 39 start_index: 68 end_index: 69
35	ExprRange	lambda_map: 40 start_index: 68 end_index: 78
36	Literal
37	Literal
38	ExprTuple	41
39	Lambda	parameter: 60 body: 42
40	Lambda	parameter: 60 body: 43
41	Lambda	parameter: 82 body: 44
42	Operation	operator: 54 operands: 45
43	Operation	operator: 71 operand: 51
44	Conditional	value: 47 condition: 48
45	ExprTuple	49, 50
46	ExprTuple	51
47	Operation	operator: 52 operands: 53
48	Operation	operator: 54 operands: 55
49	IndexedVar	variable: 61 index: 60
50	Literal
51	IndexedVar	variable: 80 index: 60
52	Literal
53	ExprTuple	57, 58
54	Literal
55	ExprTuple	82, 59
56	ExprTuple	60
57	IndexedVar	variable: 61 index: 82
58	Operation	operator: 62 operands: 63
59	Operation	operator: 64 operands: 65
60	Variable
61	Variable
62	Literal
63	ExprTuple	66, 67
64	Literal
65	ExprTuple	68, 69
66	Operation	operator: 71 operand: 75
67	Operation	operator: 71 operand: 76
68	Literal
69	Operation	operator: 73 operands: 74
70	ExprTuple	75
71	Literal
72	ExprTuple	76
73	Literal
74	ExprTuple	77, 78
75	IndexedVar	variable: 79 index: 82
76	IndexedVar	variable: 80 index: 82
77	Variable
78	Variable
79	Variable
80	Variable
81	ExprTuple	82
82	Variable