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 P, Px, Py, X, x, y
from proveit.linear_algebra import HilbertSpaces, Hspace, InnerProd, LinMap, OrthoProj, VecZero
from proveit.logic import And, Equals, Forall, Iff, Implies, SubsetEq
from proveit.numbers import zero

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Forall(instance_param_or_params = [Hspace, X], instance_expr = Forall(instance_param_or_params = [P], instance_expr = Iff(Equals(P, OrthoProj(Hspace, X)), And(Forall(instance_param_or_params = sub_expr1, instance_expr = Equals(Px, x), domain = X), Forall(instance_param_or_params = [y], instance_expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = Equals(InnerProd(x, y), zero), domain = X), Equals(Py, VecZero(Hspace))), domain = Hspace)).with_wrapping_at(2)).with_wrapping_at(2), domain = LinMap(Hspace, Hspace)), domain = HilbertSpaces, condition = SubsetEq(X, 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())

\forall_{\mathcal{H}, X \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces}~|~X \subseteq \mathcal{H}}~\left[\forall_{P \in \mathcal{L}\left(\mathcal{H}, \mathcal{H}\right)}~\left(\begin{array}{c} \begin{array}{l} \left(P = \textrm{OrthoProj}\left(\mathcal{H}, X\right)\right) \Leftrightarrow  \\ \left(\begin{array}{c} \left[\forall_{x \in X}~\left(P\left(x\right) = x\right)\right] \land  \\ \left[\forall_{y \in \mathcal{H}}~\left(\left[\forall_{x \in X}~\left(\left \langle x, y\right \rangle = 0\right)\right] \Rightarrow \left(P\left(y\right) = \vec{0}\left(\mathcal{H}\right)\right)\right)\right] \end{array}\right) \end{array} \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: 54 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 37 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 54 operand: 8
5	Operation	operator: 29 operands: 7
6	ExprTuple	8
7	ExprTuple	9, 10, 11
8	Lambda	parameter: 62 body: 13
9	Operation	operator: 15 operands: 14
10	Operation	operator: 15 operands: 16
11	Operation	operator: 17 operands: 18
12	ExprTuple	62
13	Conditional	value: 19 condition: 20
14	ExprTuple	68, 21
15	Literal
16	ExprTuple	75, 21
17	Literal
18	ExprTuple	75, 68
19	Operation	operator: 22 operands: 23
20	Operation	operator: 71 operands: 24
21	Literal
22	Literal
23	ExprTuple	25, 26
24	ExprTuple	62, 27
25	Operation	operator: 69 operands: 28
26	Operation	operator: 29 operands: 30
27	Operation	operator: 31 operands: 32
28	ExprTuple	62, 33
29	Literal
30	ExprTuple	34, 35
31	Literal
32	ExprTuple	68, 68
33	Operation	operator: 36 operands: 37
34	Operation	operator: 54 operand: 40
35	Operation	operator: 54 operand: 41
36	Literal
37	ExprTuple	68, 75
38	ExprTuple	40
39	ExprTuple	41
40	Lambda	parameter: 78 body: 42
41	Lambda	parameter: 79 body: 43
42	Conditional	value: 44 condition: 67
43	Conditional	value: 45 condition: 46
44	Operation	operator: 69 operands: 47
45	Operation	operator: 48 operands: 49
46	Operation	operator: 71 operands: 50
47	ExprTuple	51, 78
48	Literal
49	ExprTuple	52, 53
50	ExprTuple	79, 68
51	Operation	operator: 62 operand: 78
52	Operation	operator: 54 operand: 57
53	Operation	operator: 69 operands: 56
54	Literal
55	ExprTuple	57
56	ExprTuple	58, 59
57	Lambda	parameter: 78 body: 61
58	Operation	operator: 62 operand: 79
59	Operation	operator: 64 operand: 68
60	ExprTuple	78
61	Conditional	value: 66 condition: 67
62	Variable
63	ExprTuple	79
64	Literal
65	ExprTuple	68
66	Operation	operator: 69 operands: 70
67	Operation	operator: 71 operands: 72
68	Variable
69	Literal
70	ExprTuple	73, 74
71	Literal
72	ExprTuple	78, 75
73	Operation	operator: 76 operands: 77
74	Literal
75	Variable
76	Literal
77	ExprTuple	78, 79
78	Variable
79	Variable