Expression of type Conditional

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 Conditional, ExprRange, H, IndexedVar, Variable, n, v, w, x
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, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Set(x_1_to_n)
expr = Conditional(Equals(InSet(sub_expr2, OrthoNormBases(H)), And(InSet(sub_expr2, Bases(H)), Forall(instance_param_or_params = [v, w], instance_expr = Equals(InnerProd(v, w), KroneckerDelta(v, w)), domain = sub_expr2))).with_wrapping_at(2), And(ExprRange(sub_expr1, InSet(IndexedVar(x, sub_expr1), H), one, n)))

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())

\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} \textrm{ if } \left(x_{1} \in H\right) \land  \left(x_{2} \in H\right) \land  \ldots \land  \left(x_{n} \in H\right)\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

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