Expression of type Conditional

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 Conditional, P, X
from proveit.linear_algebra import HilbertSpaces, Hspace, OrthoProj
from proveit.logic import Equals, Forall, InClass, SubsetEq
from proveit.numbers import Exp, two

# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [X], instance_expr = Forall(instance_param_or_params = [P], instance_expr = Equals(Exp(P, two), P), condition = Equals(P, OrthoProj(Hspace, X))), condition = SubsetEq(X, Hspace)), InClass(Hspace, 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())

\left\{\forall_{X~|~X \subseteq \mathcal{H}}~\left[\forall_{P~|~P = \textrm{OrthoProj}\left(\mathcal{H}, X\right)}~\left(P^{2} = P\right)\right] \textrm{ if } \mathcal{H} \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces}\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

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