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 Hspace, InnerProd, LinMap, OrthoProj, VecZero
from proveit.logic import And, Equals, Forall, Iff, Implies
from proveit.numbers import zero

# build up the expression from sub-expressions
sub_expr1 = [x]
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))

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_{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)

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: 39 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 47 body: 4
3	ExprTuple	47
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operands: 8
6	Operation	operator: 56 operands: 9
7	Literal
8	ExprTuple	10, 11
9	ExprTuple	47, 12
10	Operation	operator: 54 operands: 13
11	Operation	operator: 14 operands: 15
12	Operation	operator: 16 operands: 17
13	ExprTuple	47, 18
14	Literal
15	ExprTuple	19, 20
16	Literal
17	ExprTuple	53, 53
18	Operation	operator: 21 operands: 22
19	Operation	operator: 39 operand: 25
20	Operation	operator: 39 operand: 26
21	Literal
22	ExprTuple	53, 60
23	ExprTuple	25
24	ExprTuple	26
25	Lambda	parameter: 63 body: 27
26	Lambda	parameter: 64 body: 28
27	Conditional	value: 29 condition: 52
28	Conditional	value: 30 condition: 31
29	Operation	operator: 54 operands: 32
30	Operation	operator: 33 operands: 34
31	Operation	operator: 56 operands: 35
32	ExprTuple	36, 63
33	Literal
34	ExprTuple	37, 38
35	ExprTuple	64, 53
36	Operation	operator: 47 operand: 63
37	Operation	operator: 39 operand: 42
38	Operation	operator: 54 operands: 41
39	Literal
40	ExprTuple	42
41	ExprTuple	43, 44
42	Lambda	parameter: 63 body: 46
43	Operation	operator: 47 operand: 64
44	Operation	operator: 49 operand: 53
45	ExprTuple	63
46	Conditional	value: 51 condition: 52
47	Variable
48	ExprTuple	64
49	Literal
50	ExprTuple	53
51	Operation	operator: 54 operands: 55
52	Operation	operator: 56 operands: 57
53	Variable
54	Literal
55	ExprTuple	58, 59
56	Literal
57	ExprTuple	63, 60
58	Operation	operator: 61 operands: 62
59	Literal
60	Variable
61	Literal
62	ExprTuple	63, 64
63	Variable
64	Variable