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

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Lambda([Hspace, X], Conditional(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)), And(InClass(Hspace, HilbertSpaces), InClass(X, HilbertSpaces), 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())

\left(\mathcal{H}, X\right) \mapsto \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) \textrm{ if } \mathcal{H} \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces} ,  X \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces} ,  X \subseteq \mathcal{H}\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameters: 35 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 52 operand: 6
3	Operation	operator: 27 operands: 5
4	ExprTuple	6
5	ExprTuple	7, 8, 9
6	Lambda	parameter: 60 body: 11
7	Operation	operator: 13 operands: 12
8	Operation	operator: 13 operands: 14
9	Operation	operator: 15 operands: 16
10	ExprTuple	60
11	Conditional	value: 17 condition: 18
12	ExprTuple	66, 19
13	Literal
14	ExprTuple	73, 19
15	Literal
16	ExprTuple	73, 66
17	Operation	operator: 20 operands: 21
18	Operation	operator: 69 operands: 22
19	Literal
20	Literal
21	ExprTuple	23, 24
22	ExprTuple	60, 25
23	Operation	operator: 67 operands: 26
24	Operation	operator: 27 operands: 28
25	Operation	operator: 29 operands: 30
26	ExprTuple	60, 31
27	Literal
28	ExprTuple	32, 33
29	Literal
30	ExprTuple	66, 66
31	Operation	operator: 34 operands: 35
32	Operation	operator: 52 operand: 38
33	Operation	operator: 52 operand: 39
34	Literal
35	ExprTuple	66, 73
36	ExprTuple	38
37	ExprTuple	39
38	Lambda	parameter: 76 body: 40
39	Lambda	parameter: 77 body: 41
40	Conditional	value: 42 condition: 65
41	Conditional	value: 43 condition: 44
42	Operation	operator: 67 operands: 45
43	Operation	operator: 46 operands: 47
44	Operation	operator: 69 operands: 48
45	ExprTuple	49, 76
46	Literal
47	ExprTuple	50, 51
48	ExprTuple	77, 66
49	Operation	operator: 60 operand: 76
50	Operation	operator: 52 operand: 55
51	Operation	operator: 67 operands: 54
52	Literal
53	ExprTuple	55
54	ExprTuple	56, 57
55	Lambda	parameter: 76 body: 59
56	Operation	operator: 60 operand: 77
57	Operation	operator: 62 operand: 66
58	ExprTuple	76
59	Conditional	value: 64 condition: 65
60	Variable
61	ExprTuple	77
62	Literal
63	ExprTuple	66
64	Operation	operator: 67 operands: 68
65	Operation	operator: 69 operands: 70
66	Variable
67	Literal
68	ExprTuple	71, 72
69	Literal
70	ExprTuple	76, 73
71	Operation	operator: 74 operands: 75
72	Literal
73	Variable
74	Literal
75	ExprTuple	76, 77
76	Variable
77	Variable