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

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Lambda(P, Conditional(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), InSet(P, 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())

P \mapsto \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} \textrm{ if } P \in \mathcal{L}\left(\mathcal{H}, \mathcal{H}\right)\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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