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 VecSpaces, lin_comb_axn
from proveit.logic import Forall, InSet

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [V], instance_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), domain = VecSpaces(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_{V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\left[\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]\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: 20 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 51 body: 4
3	ExprTuple	51
4	Conditional	value: 5 condition: 6
5	Operation	operator: 20 operand: 10
6	Operation	operator: 8 operands: 9
7	ExprTuple	10
8	Literal
9	ExprTuple	51, 11
10	Lambda	parameters: 12 body: 13
11	Operation	operator: 14 operand: 39
12	ExprTuple	16
13	Conditional	value: 17 condition: 18
14	Literal
15	ExprTuple	39
16	ExprRange	lambda_map: 19 start_index: 46 end_index: 47
17	Operation	operator: 20 operand: 23
18	Operation	operator: 34 operands: 22
19	Lambda	parameter: 59 body: 54
20	Literal
21	ExprTuple	23
22	ExprTuple	24
23	Lambda	parameters: 25 body: 26
24	ExprRange	lambda_map: 27 start_index: 46 end_index: 47
25	ExprTuple	28
26	Conditional	value: 29 condition: 30
27	Lambda	parameter: 59 body: 31
28	ExprRange	lambda_map: 32 start_index: 46 end_index: 47
29	Operation	operator: 48 operands: 33
30	Operation	operator: 34 operands: 35
31	Operation	operator: 48 operands: 36
32	Lambda	parameter: 59 body: 55
33	ExprTuple	37, 51
34	Literal
35	ExprTuple	38
36	ExprTuple	54, 39
37	Operation	operator: 40 operands: 41
38	ExprRange	lambda_map: 42 start_index: 46 end_index: 47
39	Variable
40	Literal
41	ExprTuple	43
42	Lambda	parameter: 59 body: 44
43	ExprRange	lambda_map: 45 start_index: 46 end_index: 47
44	Operation	operator: 48 operands: 49
45	Lambda	parameter: 59 body: 50
46	Literal
47	Variable
48	Literal
49	ExprTuple	55, 51
50	Operation	operator: 52 operands: 53
51	Variable
52	Literal
53	ExprTuple	54, 55
54	IndexedVar	variable: 56 index: 59
55	IndexedVar	variable: 57 index: 59
56	Variable
57	Variable
58	ExprTuple	59
59	Variable