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, H, K, Lambda, n, v, w
from proveit.core_expr_types import x_1_to_n
from proveit.linear_algebra import Bases, InnerProd, InnerProdSpaces, OrthoNormBases
from proveit.logic import And, Equals, Forall, InClass, InSet, Set
from proveit.numbers import KroneckerDelta, NaturalPos

# build up the expression from sub-expressions
sub_expr1 = Set(x_1_to_n)
expr = Lambda(H, Conditional(Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [x_1_to_n], instance_expr = Equals(InSet(sub_expr1, OrthoNormBases(H)), And(InSet(sub_expr1, Bases(H)), Forall(instance_param_or_params = [v, w], instance_expr = Equals(InnerProd(v, w), KroneckerDelta(v, w)), domain = sub_expr1))).with_wrapping_at(2), domain = H), domain = NaturalPos), InClass(H, InnerProdSpaces(K))))

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())

H \mapsto \left\{\forall_{n \in \mathbb{N}^+}~\left[\forall_{x_{1}, x_{2}, \ldots, x_{n} \in H}~\left(\begin{array}{c} \begin{array}{l} \left(\left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \in \textrm{O.N.Bases}\left(H\right)\right) =  \\ \left(\left(\left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \in \textrm{Bases}\left(H\right)\right) \land \left[\forall_{v, w \in \left\{x_{1}, x_{2}, \ldots, x_{n}\right\}}~\left(\left \langle v, w\right \rangle = \delta_{v, w}\right)\right]\right) \end{array} \end{array}\right)\right] \textrm{ if } H \underset{{\scriptscriptstyle c}}{\in} \textrm{InnerProdSpaces}\left(K\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: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 37 operand: 7
3	Operation	operator: 5 operands: 6
4	ExprTuple	7
5	Literal
6	ExprTuple	45, 8
7	Lambda	parameter: 70 body: 10
8	Operation	operator: 11 operand: 15
9	ExprTuple	70
10	Conditional	value: 13 condition: 14
11	Literal
12	ExprTuple	15
13	Operation	operator: 37 operand: 18
14	Operation	operator: 60 operands: 17
15	Variable
16	ExprTuple	18
17	ExprTuple	70, 19
18	Lambda	parameters: 66 body: 20
19	Literal
20	Conditional	value: 21 condition: 22
21	Operation	operator: 48 operands: 23
22	Operation	operator: 50 operands: 24
23	ExprTuple	25, 26
24	ExprTuple	27
25	Operation	operator: 60 operands: 28
26	Operation	operator: 50 operands: 29
27	ExprRange	lambda_map: 30 start_index: 69 end_index: 70
28	ExprTuple	64, 31
29	ExprTuple	32, 33
30	Lambda	parameter: 74 body: 34
31	Operation	operator: 35 operand: 45
32	Operation	operator: 60 operands: 36
33	Operation	operator: 37 operand: 41
34	Operation	operator: 60 operands: 39
35	Literal
36	ExprTuple	64, 40
37	Literal
38	ExprTuple	41
39	ExprTuple	71, 45
40	Operation	operator: 42 operand: 45
41	Lambda	parameters: 58 body: 44
42	Literal
43	ExprTuple	45
44	Conditional	value: 46 condition: 47
45	Variable
46	Operation	operator: 48 operands: 49
47	Operation	operator: 50 operands: 51
48	Literal
49	ExprTuple	52, 53
50	Literal
51	ExprTuple	54, 55
52	Operation	operator: 56 operands: 58
53	Operation	operator: 57 operands: 58
54	Operation	operator: 60 operands: 59
55	Operation	operator: 60 operands: 61
56	Literal
57	Literal
58	ExprTuple	62, 63
59	ExprTuple	62, 64
60	Literal
61	ExprTuple	63, 64
62	Variable
63	Variable
64	Operation	operator: 65 operands: 66
65	Literal
66	ExprTuple	67
67	ExprRange	lambda_map: 68 start_index: 69 end_index: 70
68	Lambda	parameter: 74 body: 71
69	Literal
70	Variable
71	IndexedVar	variable: 72 index: 74
72	Variable
73	ExprTuple	74
74	Variable