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 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, InSet, Set
from proveit.numbers import KroneckerDelta, NaturalPos

# build up the expression from sub-expressions
sub_expr1 = Set(x_1_to_n)
expr = Lambda(K, Forall(instance_param_or_params = [H], instance_expr = 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), domain = 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())

K \mapsto \left[\forall_{H \underset{{\scriptscriptstyle c}}{\in} \textrm{InnerProdSpaces}\left(K\right)}~\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]\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: 18 body: 1
1	Operation	operator: 40 operand: 3
2	ExprTuple	3
3	Lambda	parameter: 48 body: 4
4	Conditional	value: 5 condition: 6
5	Operation	operator: 40 operand: 10
6	Operation	operator: 8 operands: 9
7	ExprTuple	10
8	Literal
9	ExprTuple	48, 11
10	Lambda	parameter: 73 body: 13
11	Operation	operator: 14 operand: 18
12	ExprTuple	73
13	Conditional	value: 16 condition: 17
14	Literal
15	ExprTuple	18
16	Operation	operator: 40 operand: 21
17	Operation	operator: 63 operands: 20
18	Variable
19	ExprTuple	21
20	ExprTuple	73, 22
21	Lambda	parameters: 69 body: 23
22	Literal
23	Conditional	value: 24 condition: 25
24	Operation	operator: 51 operands: 26
25	Operation	operator: 53 operands: 27
26	ExprTuple	28, 29
27	ExprTuple	30
28	Operation	operator: 63 operands: 31
29	Operation	operator: 53 operands: 32
30	ExprRange	lambda_map: 33 start_index: 72 end_index: 73
31	ExprTuple	67, 34
32	ExprTuple	35, 36
33	Lambda	parameter: 77 body: 37
34	Operation	operator: 38 operand: 48
35	Operation	operator: 63 operands: 39
36	Operation	operator: 40 operand: 44
37	Operation	operator: 63 operands: 42
38	Literal
39	ExprTuple	67, 43
40	Literal
41	ExprTuple	44
42	ExprTuple	74, 48
43	Operation	operator: 45 operand: 48
44	Lambda	parameters: 61 body: 47
45	Literal
46	ExprTuple	48
47	Conditional	value: 49 condition: 50
48	Variable
49	Operation	operator: 51 operands: 52
50	Operation	operator: 53 operands: 54
51	Literal
52	ExprTuple	55, 56
53	Literal
54	ExprTuple	57, 58
55	Operation	operator: 59 operands: 61
56	Operation	operator: 60 operands: 61
57	Operation	operator: 63 operands: 62
58	Operation	operator: 63 operands: 64
59	Literal
60	Literal
61	ExprTuple	65, 66
62	ExprTuple	65, 67
63	Literal
64	ExprTuple	66, 67
65	Variable
66	Variable
67	Operation	operator: 68 operands: 69
68	Literal
69	ExprTuple	70
70	ExprRange	lambda_map: 71 start_index: 72 end_index: 73
71	Lambda	parameter: 77 body: 74
72	Literal
73	Variable
74	IndexedVar	variable: 75 index: 77
75	Variable
76	ExprTuple	77
77	Variable