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
from proveit.linear_algebra import InnerProd, InnerProdSpaces, Norm
from proveit.logic import Equals, Forall
from proveit.numbers import sqrt

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [K], instance_expr = Forall(instance_param_or_params = [H], instance_expr = Forall(instance_param_or_params = [x], instance_expr = Equals(Norm(x), sqrt(InnerProd(x, x))), domain = H), domain = InnerProdSpaces(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}~\left[\forall_{H \underset{{\scriptscriptstyle c}}{\in} \textrm{InnerProdSpaces}\left(K\right)}~\left[\forall_{x \in H}~\left(\left \|x\right \| = \sqrt{\left \langle x, x\right \rangle}\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: 10 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 21 body: 3
3	Operation	operator: 10 operand: 5
4	ExprTuple	5
5	Lambda	parameter: 28 body: 7
6	ExprTuple	28
7	Conditional	value: 8 condition: 9
8	Operation	operator: 10 operand: 14
9	Operation	operator: 12 operands: 13
10	Literal
11	ExprTuple	14
12	Literal
13	ExprTuple	28, 15
14	Lambda	parameter: 39 body: 16
15	Operation	operator: 17 operand: 21
16	Conditional	value: 19 condition: 20
17	Literal
18	ExprTuple	21
19	Operation	operator: 22 operands: 23
20	Operation	operator: 24 operands: 25
21	Variable
22	Literal
23	ExprTuple	26, 27
24	Literal
25	ExprTuple	39, 28
26	Operation	operator: 29 operand: 39
27	Operation	operator: 31 operands: 32
28	Variable
29	Literal
30	ExprTuple	39
31	Literal
32	ExprTuple	33, 34
33	Operation	operator: 35 operands: 36
34	Operation	operator: 37 operands: 38
35	Literal
36	ExprTuple	39, 39
37	Literal
38	ExprTuple	40, 41
39	Variable
40	Literal
41	Literal