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, HilbertSpaces, 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 = [Hspace], instance_expr = Forall(instance_param_or_params = [A], instance_expr = Exists(instance_param_or_params = [U], instance_expr = Equals(A, Composition(U, sqrt(Composition(Adj(A), A)))), domain = sub_expr1, condition = Equals(Composition(Adj(U), U), Identity(Hspace))).with_wrapping(), domain = sub_expr1), domain = HilbertSpaces)

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_{\mathcal{H} \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces}}~\left[\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(U \circ \sqrt{\left(A^{\dagger} \circ A\right)}\right)\right)\end{array}\right]\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: 6 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 47 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 6 operand: 10
5	Operation	operator: 8 operands: 9
6	Literal
7	ExprTuple	10
8	Literal
9	ExprTuple	47, 11
10	Lambda	parameter: 59 body: 12
11	Literal
12	Conditional	value: 13 condition: 14
13	Operation	operator: 15 operand: 18
14	Operation	operator: 29 operands: 17
15	Literal
16	ExprTuple	18
17	ExprTuple	59, 34
18	Lambda	parameter: 56 body: 19
19	Conditional	value: 20 condition: 21
20	Operation	operator: 31 operands: 22
21	Operation	operator: 23 operands: 24
22	ExprTuple	59, 25
23	Literal
24	ExprTuple	26, 27
25	Operation	operator: 48 operands: 28
26	Operation	operator: 29 operands: 30
27	Operation	operator: 31 operands: 32
28	ExprTuple	56, 33
29	Literal
30	ExprTuple	56, 34
31	Literal
32	ExprTuple	35, 36
33	Operation	operator: 37 operands: 38
34	Operation	operator: 39 operands: 40
35	Operation	operator: 48 operands: 41
36	Operation	operator: 42 operand: 47
37	Literal
38	ExprTuple	44, 45
39	Literal
40	ExprTuple	47, 47
41	ExprTuple	46, 56
42	Literal
43	ExprTuple	47
44	Operation	operator: 48 operands: 49
45	Operation	operator: 50 operands: 51
46	Operation	operator: 57 operand: 56
47	Variable
48	Literal
49	ExprTuple	53, 59
50	Literal
51	ExprTuple	54, 55
52	ExprTuple	56
53	Operation	operator: 57 operand: 59
54	Literal
55	Literal
56	Variable
57	Literal
58	ExprTuple	59
59	Variable