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 A, Composition, Conditional, Lambda, U
from proveit.linear_algebra import Adj, Hspace, Identity, LinMap
from proveit.logic import Equals, Exists, InSet
from proveit.numbers import sqrt

# build up the expression from sub-expressions
sub_expr1 = LinMap(Hspace, Hspace)
expr = Lambda(A, Conditional(Exists(instance_param_or_params = [U], instance_expr = Equals(A, Composition(sqrt(Composition(A, Adj(A))), U)), domain = sub_expr1, condition = Equals(Composition(Adj(U), U), Identity(Hspace))).with_wrapping(), InSet(A, sub_expr1)))

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())

A \mapsto \left\{\begin{array}{l}\exists_{U \in \mathcal{L}\left(\mathcal{H}, \mathcal{H}\right)~|~\left(U^{\dagger} \circ U\right) = I_{\mathcal{H}}}~\\
\left(A = \left(\sqrt{\left(A \circ A^{\dagger}\right)} \circ U\right)\right)\end{array} \textrm{ if } A \in \mathcal{L}\left(\mathcal{H}, \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	parameter: 48 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 4 operand: 7
3	Operation	operator: 18 operands: 6
4	Literal
5	ExprTuple	7
6	ExprTuple	48, 23
7	Lambda	parameter: 45 body: 8
8	Conditional	value: 9 condition: 10
9	Operation	operator: 20 operands: 11
10	Operation	operator: 12 operands: 13
11	ExprTuple	48, 14
12	Literal
13	ExprTuple	15, 16
14	Operation	operator: 37 operands: 17
15	Operation	operator: 18 operands: 19
16	Operation	operator: 20 operands: 21
17	ExprTuple	22, 45
18	Literal
19	ExprTuple	45, 23
20	Literal
21	ExprTuple	24, 25
22	Operation	operator: 26 operands: 27
23	Operation	operator: 28 operands: 29
24	Operation	operator: 37 operands: 30
25	Operation	operator: 31 operand: 36
26	Literal
27	ExprTuple	33, 34
28	Literal
29	ExprTuple	36, 36
30	ExprTuple	35, 45
31	Literal
32	ExprTuple	36
33	Operation	operator: 37 operands: 38
34	Operation	operator: 39 operands: 40
35	Operation	operator: 46 operand: 45
36	Variable
37	Literal
38	ExprTuple	48, 42
39	Literal
40	ExprTuple	43, 44
41	ExprTuple	45
42	Operation	operator: 46 operand: 48
43	Literal
44	Literal
45	Variable
46	Literal
47	ExprTuple	48
48	Variable