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, n, v, w
from proveit.core_expr_types import x_1_to_n
from proveit.linear_algebra import Bases, InnerProd, 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 = [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)

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_{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]

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