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 = [K], instance_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_{K}~\left[\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]\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: 23 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 42 body: 3
3	Operation	operator: 23 operand: 5
4	ExprTuple	5
5	Lambda	parameter: 54 body: 7
6	ExprTuple	54
7	Conditional	value: 8 condition: 9
8	Operation	operator: 23 operand: 13
9	Operation	operator: 11 operands: 12
10	ExprTuple	13
11	Literal
12	ExprTuple	54, 14
13	Lambda	parameters: 15 body: 16
14	Operation	operator: 17 operand: 42
15	ExprTuple	19
16	Conditional	value: 20 condition: 21
17	Literal
18	ExprTuple	42
19	ExprRange	lambda_map: 22 start_index: 49 end_index: 50
20	Operation	operator: 23 operand: 26
21	Operation	operator: 37 operands: 25
22	Lambda	parameter: 62 body: 57
23	Literal
24	ExprTuple	26
25	ExprTuple	27
26	Lambda	parameters: 28 body: 29
27	ExprRange	lambda_map: 30 start_index: 49 end_index: 50
28	ExprTuple	31
29	Conditional	value: 32 condition: 33
30	Lambda	parameter: 62 body: 34
31	ExprRange	lambda_map: 35 start_index: 49 end_index: 50
32	Operation	operator: 51 operands: 36
33	Operation	operator: 37 operands: 38
34	Operation	operator: 51 operands: 39
35	Lambda	parameter: 62 body: 58
36	ExprTuple	40, 54
37	Literal
38	ExprTuple	41
39	ExprTuple	57, 42
40	Operation	operator: 43 operands: 44
41	ExprRange	lambda_map: 45 start_index: 49 end_index: 50
42	Variable
43	Literal
44	ExprTuple	46
45	Lambda	parameter: 62 body: 47
46	ExprRange	lambda_map: 48 start_index: 49 end_index: 50
47	Operation	operator: 51 operands: 52
48	Lambda	parameter: 62 body: 53
49	Literal
50	Variable
51	Literal
52	ExprTuple	58, 54
53	Operation	operator: 55 operands: 56
54	Variable
55	Literal
56	ExprTuple	57, 58
57	IndexedVar	variable: 59 index: 62
58	IndexedVar	variable: 60 index: 62
59	Variable
60	Variable
61	ExprTuple	62
62	Variable