Expression of type Forall

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, Forall
from proveit.numbers import sqrt

# build up the expression from sub-expressions
sub_expr1 = LinMap(Hspace, Hspace)
expr = Forall(instance_param_or_params = [A], instance_expr = 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(), domain = 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())

\forall_{A \in \mathcal{L}\left(\mathcal{H}, \mathcal{H}\right)}~\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}\right]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('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: 51 body: 4
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 21 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	51, 26
10	Lambda	parameter: 48 body: 11
11	Conditional	value: 12 condition: 13
12	Operation	operator: 23 operands: 14
13	Operation	operator: 15 operands: 16
14	ExprTuple	51, 17
15	Literal
16	ExprTuple	18, 19
17	Operation	operator: 40 operands: 20
18	Operation	operator: 21 operands: 22
19	Operation	operator: 23 operands: 24
20	ExprTuple	25, 48
21	Literal
22	ExprTuple	48, 26
23	Literal
24	ExprTuple	27, 28
25	Operation	operator: 29 operands: 30
26	Operation	operator: 31 operands: 32
27	Operation	operator: 40 operands: 33
28	Operation	operator: 34 operand: 39
29	Literal
30	ExprTuple	36, 37
31	Literal
32	ExprTuple	39, 39
33	ExprTuple	38, 48
34	Literal
35	ExprTuple	39
36	Operation	operator: 40 operands: 41
37	Operation	operator: 42 operands: 43
38	Operation	operator: 49 operand: 48
39	Variable
40	Literal
41	ExprTuple	51, 45
42	Literal
43	ExprTuple	46, 47
44	ExprTuple	48
45	Operation	operator: 49 operand: 51
46	Literal
47	Literal
48	Variable
49	Literal
50	ExprTuple	51
51	Variable