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, Py, X, x, y
from proveit.linear_algebra import Hspace, InnerProd, VecZero
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import zero

# build up the expression from sub-expressions
expr = Conditional(Implies(Forall(instance_param_or_params = [x], instance_expr = Equals(InnerProd(x, y), zero), domain = X), Equals(Py, VecZero(Hspace))), InSet(y, Hspace))

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\{\left[\forall_{x \in X}~\left(\left \langle x, y\right \rangle = 0\right)\right] \Rightarrow \left(P\left(y\right) = \vec{0}\left(\mathcal{H}\right)\right) \textrm{ if } y \in \mathcal{H}\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: 3 operands: 4
2	Operation	operator: 25 operands: 5
3	Literal
4	ExprTuple	6, 7
5	ExprTuple	33, 22
6	Operation	operator: 8 operand: 11
7	Operation	operator: 23 operands: 10
8	Literal
9	ExprTuple	11
10	ExprTuple	12, 13
11	Lambda	parameter: 32 body: 15
12	Operation	operator: 16 operand: 33
13	Operation	operator: 18 operand: 22
14	ExprTuple	32
15	Conditional	value: 20 condition: 21
16	Variable
17	ExprTuple	33
18	Literal
19	ExprTuple	22
20	Operation	operator: 23 operands: 24
21	Operation	operator: 25 operands: 26
22	Variable
23	Literal
24	ExprTuple	27, 28
25	Literal
26	ExprTuple	32, 29
27	Operation	operator: 30 operands: 31
28	Literal
29	Variable
30	Literal
31	ExprTuple	32, 33
32	Variable
33	Variable