Expression of type Exists

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, U
from proveit.linear_algebra import Adj, Hspace, Identity, LinMap
from proveit.logic import Equals, Exists
from proveit.numbers import sqrt

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

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())

\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(U \circ \sqrt{\left(A^{\dagger} \circ A\right)}\right)\right)\end{array}

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	True	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameter: 41 body: 4
4	Conditional	value: 5 condition: 6
5	Operation	operator: 16 operands: 7
6	Operation	operator: 8 operands: 9
7	ExprTuple	44, 10
8	Literal
9	ExprTuple	11, 12
10	Operation	operator: 33 operands: 13
11	Operation	operator: 14 operands: 15
12	Operation	operator: 16 operands: 17
13	ExprTuple	41, 18
14	Literal
15	ExprTuple	41, 19
16	Literal
17	ExprTuple	20, 21
18	Operation	operator: 22 operands: 23
19	Operation	operator: 24 operands: 25
20	Operation	operator: 33 operands: 26
21	Operation	operator: 27 operand: 32
22	Literal
23	ExprTuple	29, 30
24	Literal
25	ExprTuple	32, 32
26	ExprTuple	31, 41
27	Literal
28	ExprTuple	32
29	Operation	operator: 33 operands: 34
30	Operation	operator: 35 operands: 36
31	Operation	operator: 42 operand: 41
32	Variable
33	Literal
34	ExprTuple	38, 44
35	Literal
36	ExprTuple	39, 40
37	ExprTuple	41
38	Operation	operator: 42 operand: 44
39	Literal
40	Literal
41	Variable
42	Literal
43	ExprTuple	44
44	Variable