Expression of type Lambda

from the theory of proveit.linear_algebra.inner_products

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, Lambda, M, i, n
from proveit.core_expr_types import a_1_to_n, a_i, x_1_to_n, x_i
from proveit.linear_algebra import Hspace, OrthoNormBases, OrthoProj, ScalarMult, Span, VecSum
from proveit.logic import Equals, Exists, InSet, Set
from proveit.numbers import Complex, Interval, one

# build up the expression from sub-expressions
expr = Lambda([x_1_to_n], Conditional(Exists(instance_param_or_params = [a_1_to_n], instance_expr = Equals(M, VecSum(index_or_indices = [i], summand = ScalarMult(a_i, OrthoProj(Hspace, Span(Set(x_i)))), domain = Interval(one, n))), domain = Complex).with_wrapping(), InSet(Set(x_1_to_n), OrthoNormBases(Hspace))))

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(x_{1}, x_{2}, \ldots, x_{n}\right) \mapsto \left\{\begin{array}{l}\exists_{a_{1}, a_{2}, \ldots, a_{n} \in \mathbb{C}}~\\
\left(M = \left(\sum_{i=1}^{n} \left(a_{i} \cdot \textrm{OrthoProj}\left(\mathcal{H}, \textrm{Span}\left(\left\{x_{i}\right\}\right)\right)\right)\right)\right)\end{array} \textrm{ if } \left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \in \textrm{O.N.Bases}\left(\mathcal{H}\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: 12 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 4 operand: 7
3	Operation	operator: 42 operands: 6
4	Literal
5	ExprTuple	7
6	ExprTuple	8, 9
7	Lambda	parameters: 10 body: 11
8	Operation	operator: 61 operands: 12
9	Operation	operator: 13 operand: 54
10	ExprTuple	15
11	Conditional	value: 16 condition: 17
12	ExprTuple	18
13	Literal
14	ExprTuple	54
15	ExprRange	lambda_map: 19 start_index: 56 end_index: 57
16	Operation	operator: 20 operands: 21
17	Operation	operator: 22 operands: 23
18	ExprRange	lambda_map: 24 start_index: 56 end_index: 57
19	Lambda	parameter: 48 body: 38
20	Literal
21	ExprTuple	25, 26
22	Literal
23	ExprTuple	27
24	Lambda	parameter: 48 body: 28
25	Variable
26	Operation	operator: 29 operand: 32
27	ExprRange	lambda_map: 31 start_index: 56 end_index: 57
28	IndexedVar	variable: 64 index: 48
29	Literal
30	ExprTuple	32
31	Lambda	parameter: 48 body: 33
32	Lambda	parameter: 66 body: 34
33	Operation	operator: 42 operands: 35
34	Conditional	value: 36 condition: 37
35	ExprTuple	38, 39
36	Operation	operator: 40 operands: 41
37	Operation	operator: 42 operands: 43
38	IndexedVar	variable: 49 index: 48
39	Literal
40	Literal
41	ExprTuple	45, 46
42	Literal
43	ExprTuple	66, 47
44	ExprTuple	48
45	IndexedVar	variable: 49 index: 66
46	Operation	operator: 50 operands: 51
47	Operation	operator: 52 operands: 53
48	Variable
49	Variable
50	Literal
51	ExprTuple	54, 55
52	Literal
53	ExprTuple	56, 57
54	Variable
55	Operation	operator: 58 operand: 60
56	Literal
57	Variable
58	Literal
59	ExprTuple	60
60	Operation	operator: 61 operand: 63
61	Literal
62	ExprTuple	63
63	IndexedVar	variable: 64 index: 66
64	Variable
65	ExprTuple	66
66	Variable