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, n
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
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [n], instance_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))), domain = NaturalPos)

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_{n \in \mathbb{N}^+}~\left[\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]\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: 31 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 58 body: 4
3	ExprTuple	58
4	Conditional	value: 5 condition: 6
5	Operation	operator: 31 operand: 9
6	Operation	operator: 59 operands: 8
7	ExprTuple	9
8	ExprTuple	58, 10
9	Lambda	parameter: 50 body: 11
10	Literal
11	Operation	operator: 31 operand: 13
12	ExprTuple	13
13	Lambda	parameter: 62 body: 15
14	ExprTuple	62
15	Conditional	value: 16 condition: 17
16	Operation	operator: 31 operand: 21
17	Operation	operator: 19 operands: 20
18	ExprTuple	21
19	Literal
20	ExprTuple	62, 22
21	Lambda	parameters: 23 body: 24
22	Operation	operator: 25 operand: 50
23	ExprTuple	27
24	Conditional	value: 28 condition: 29
25	Literal
26	ExprTuple	50
27	ExprRange	lambda_map: 30 start_index: 57 end_index: 58
28	Operation	operator: 31 operand: 34
29	Operation	operator: 45 operands: 33
30	Lambda	parameter: 70 body: 65
31	Literal
32	ExprTuple	34
33	ExprTuple	35
34	Lambda	parameters: 36 body: 37
35	ExprRange	lambda_map: 38 start_index: 57 end_index: 58
36	ExprTuple	39
37	Conditional	value: 40 condition: 41
38	Lambda	parameter: 70 body: 42
39	ExprRange	lambda_map: 43 start_index: 57 end_index: 58
40	Operation	operator: 59 operands: 44
41	Operation	operator: 45 operands: 46
42	Operation	operator: 59 operands: 47
43	Lambda	parameter: 70 body: 66
44	ExprTuple	48, 62
45	Literal
46	ExprTuple	49
47	ExprTuple	65, 50
48	Operation	operator: 51 operands: 52
49	ExprRange	lambda_map: 53 start_index: 57 end_index: 58
50	Variable
51	Literal
52	ExprTuple	54
53	Lambda	parameter: 70 body: 55
54	ExprRange	lambda_map: 56 start_index: 57 end_index: 58
55	Operation	operator: 59 operands: 60
56	Lambda	parameter: 70 body: 61
57	Literal
58	Variable
59	Literal
60	ExprTuple	66, 62
61	Operation	operator: 63 operands: 64
62	Variable
63	Literal
64	ExprTuple	65, 66
65	IndexedVar	variable: 67 index: 70
66	IndexedVar	variable: 68 index: 70
67	Variable
68	Variable
69	ExprTuple	70
70	Variable