Expression of type Lambda

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, Lambda, 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 = Lambda(Hspace, 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())

\mathcal{H} \mapsto \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()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 34 body: 2
1	ExprTuple	34
2	Conditional	value: 3 condition: 4
3	Operation	operator: 14 operand: 8
4	Operation	operator: 6 operands: 7
5	ExprTuple	8
6	Literal
7	ExprTuple	34, 9
8	Lambda	parameter: 35 body: 11
9	Literal
10	ExprTuple	35
11	Conditional	value: 12 condition: 13
12	Operation	operator: 14 operand: 18
13	Operation	operator: 16 operands: 17
14	Literal
15	ExprTuple	18
16	Literal
17	ExprTuple	35, 34
18	Lambda	parameter: 32 body: 20
19	ExprTuple	32
20	Conditional	value: 21 condition: 22
21	Operation	operator: 24 operands: 23
22	Operation	operator: 24 operands: 25
23	ExprTuple	26, 32
24	Literal
25	ExprTuple	32, 27
26	Operation	operator: 28 operands: 29
27	Operation	operator: 30 operands: 31
28	Literal
29	ExprTuple	32, 33
30	Literal
31	ExprTuple	34, 35
32	Variable
33	Literal
34	Variable
35	Variable