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, K, x, y
from proveit.linear_algebra import InnerProd, InnerProdSpaces, VecSpaces
from proveit.logic import And, Equals, Forall, InClass, InSet

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [K, H], instance_expr = Equals(InClass(H, InnerProdSpaces(K)), And(InClass(H, VecSpaces(K)), Forall(instance_param_or_params = [x, y], instance_expr = InSet(InnerProd(x, y), K)))))

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_{K, H}~\left(\left(H \underset{{\scriptscriptstyle c}}{\in} \textrm{InnerProdSpaces}\left(K\right)\right) = \left(\left(H \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)\right) \land \left[\forall_{x, y}~\left(\left \langle x, y\right \rangle \in K\right)\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: 18 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	29, 20
4	Operation	operator: 5 operands: 6
5	Literal
6	ExprTuple	7, 8
7	Operation	operator: 16 operands: 9
8	Operation	operator: 10 operands: 11
9	ExprTuple	20, 12
10	Literal
11	ExprTuple	13, 14
12	Operation	operator: 15 operand: 29
13	Operation	operator: 16 operands: 17
14	Operation	operator: 18 operand: 22
15	Literal
16	Literal
17	ExprTuple	20, 21
18	Literal
19	ExprTuple	22
20	Variable
21	Operation	operator: 23 operand: 29
22	Lambda	parameters: 31 body: 25
23	Literal
24	ExprTuple	29
25	Operation	operator: 26 operands: 27
26	Literal
27	ExprTuple	28, 29
28	Operation	operator: 30 operands: 31
29	Variable
30	Literal
31	ExprTuple	32, 33
32	Variable
33	Variable