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

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

\forall_{y \in \mathcal{H}}~\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)\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: 11 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 36 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 6 operands: 7
5	Operation	operator: 28 operands: 8
6	Literal
7	ExprTuple	9, 10
8	ExprTuple	36, 25
9	Operation	operator: 11 operand: 14
10	Operation	operator: 26 operands: 13
11	Literal
12	ExprTuple	14
13	ExprTuple	15, 16
14	Lambda	parameter: 35 body: 18
15	Operation	operator: 19 operand: 36
16	Operation	operator: 21 operand: 25
17	ExprTuple	35
18	Conditional	value: 23 condition: 24
19	Variable
20	ExprTuple	36
21	Literal
22	ExprTuple	25
23	Operation	operator: 26 operands: 27
24	Operation	operator: 28 operands: 29
25	Variable
26	Literal
27	ExprTuple	30, 31
28	Literal
29	ExprTuple	35, 32
30	Operation	operator: 33 operands: 34
31	Literal
32	Variable
33	Literal
34	ExprTuple	35, 36
35	Variable
36	Variable