Expression of type Forall

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, 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 = Forall(instance_param_or_params = [K], instance_expr = 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())

\forall_{K}~\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()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 42 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 20 body: 3
3	Operation	operator: 42 operand: 5
4	ExprTuple	5
5	Lambda	parameter: 50 body: 6
6	Conditional	value: 7 condition: 8
7	Operation	operator: 42 operand: 12
8	Operation	operator: 10 operands: 11
9	ExprTuple	12
10	Literal
11	ExprTuple	50, 13
12	Lambda	parameter: 75 body: 15
13	Operation	operator: 16 operand: 20
14	ExprTuple	75
15	Conditional	value: 18 condition: 19
16	Literal
17	ExprTuple	20
18	Operation	operator: 42 operand: 23
19	Operation	operator: 65 operands: 22
20	Variable
21	ExprTuple	23
22	ExprTuple	75, 24
23	Lambda	parameters: 71 body: 25
24	Literal
25	Conditional	value: 26 condition: 27
26	Operation	operator: 53 operands: 28
27	Operation	operator: 55 operands: 29
28	ExprTuple	30, 31
29	ExprTuple	32
30	Operation	operator: 65 operands: 33
31	Operation	operator: 55 operands: 34
32	ExprRange	lambda_map: 35 start_index: 74 end_index: 75
33	ExprTuple	69, 36
34	ExprTuple	37, 38
35	Lambda	parameter: 79 body: 39
36	Operation	operator: 40 operand: 50
37	Operation	operator: 65 operands: 41
38	Operation	operator: 42 operand: 46
39	Operation	operator: 65 operands: 44
40	Literal
41	ExprTuple	69, 45
42	Literal
43	ExprTuple	46
44	ExprTuple	76, 50
45	Operation	operator: 47 operand: 50
46	Lambda	parameters: 63 body: 49
47	Literal
48	ExprTuple	50
49	Conditional	value: 51 condition: 52
50	Variable
51	Operation	operator: 53 operands: 54
52	Operation	operator: 55 operands: 56
53	Literal
54	ExprTuple	57, 58
55	Literal
56	ExprTuple	59, 60
57	Operation	operator: 61 operands: 63
58	Operation	operator: 62 operands: 63
59	Operation	operator: 65 operands: 64
60	Operation	operator: 65 operands: 66
61	Literal
62	Literal
63	ExprTuple	67, 68
64	ExprTuple	67, 69
65	Literal
66	ExprTuple	68, 69
67	Variable
68	Variable
69	Operation	operator: 70 operands: 71
70	Literal
71	ExprTuple	72
72	ExprRange	lambda_map: 73 start_index: 74 end_index: 75
73	Lambda	parameter: 79 body: 76
74	Literal
75	Variable
76	IndexedVar	variable: 77 index: 79
77	Variable
78	ExprTuple	79
79	Variable