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, 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

# build up the expression from sub-expressions
sub_expr1 = Set(x_1_to_n)
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)

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_{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)

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