Expression of type Forall

from the theory of proveit.linear_algebra.scalar_multiplication

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 K, V
from proveit.core_expr_types import a_1_to_n, x_1_to_n
from proveit.linear_algebra import lin_comb_axn
from proveit.logic import Forall, InSet

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [a_1_to_n], instance_expr = Forall(instance_param_or_params = [x_1_to_n], instance_expr = InSet(lin_comb_axn, V), domain = V), domain = 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_{a_{1}, a_{2}, \ldots, a_{n} \in K}~\left[\forall_{x_{1}, x_{2}, \ldots, x_{n} \in V}~\left(\left(\left(a_{1} \cdot x_{1}\right) +  \left(a_{2} \cdot x_{2}\right) +  \ldots +  \left(a_{n} \cdot x_{n}\right)\right) \in V\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: 9 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	5
4	Conditional	value: 6 condition: 7
5	ExprRange	lambda_map: 8 start_index: 35 end_index: 36
6	Operation	operator: 9 operand: 12
7	Operation	operator: 23 operands: 11
8	Lambda	parameter: 48 body: 43
9	Literal
10	ExprTuple	12
11	ExprTuple	13
12	Lambda	parameters: 14 body: 15
13	ExprRange	lambda_map: 16 start_index: 35 end_index: 36
14	ExprTuple	17
15	Conditional	value: 18 condition: 19
16	Lambda	parameter: 48 body: 20
17	ExprRange	lambda_map: 21 start_index: 35 end_index: 36
18	Operation	operator: 37 operands: 22
19	Operation	operator: 23 operands: 24
20	Operation	operator: 37 operands: 25
21	Lambda	parameter: 48 body: 44
22	ExprTuple	26, 40
23	Literal
24	ExprTuple	27
25	ExprTuple	43, 28
26	Operation	operator: 29 operands: 30
27	ExprRange	lambda_map: 31 start_index: 35 end_index: 36
28	Variable
29	Literal
30	ExprTuple	32
31	Lambda	parameter: 48 body: 33
32	ExprRange	lambda_map: 34 start_index: 35 end_index: 36
33	Operation	operator: 37 operands: 38
34	Lambda	parameter: 48 body: 39
35	Literal
36	Variable
37	Literal
38	ExprTuple	44, 40
39	Operation	operator: 41 operands: 42
40	Variable
41	Literal
42	ExprTuple	43, 44
43	IndexedVar	variable: 45 index: 48
44	IndexedVar	variable: 46 index: 48
45	Variable
46	Variable
47	ExprTuple	48
48	Variable