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 = [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_{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]

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