Expression of type Conditional

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 Conditional, 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 = Conditional(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))), InSet(n, 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())

\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] \textrm{ if } n \in \mathbb{N}^+\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 27 operand: 5
2	Operation	operator: 55 operands: 4
3	ExprTuple	5
4	ExprTuple	54, 6
5	Lambda	parameter: 46 body: 7
6	Literal
7	Operation	operator: 27 operand: 9
8	ExprTuple	9
9	Lambda	parameter: 58 body: 11
10	ExprTuple	58
11	Conditional	value: 12 condition: 13
12	Operation	operator: 27 operand: 17
13	Operation	operator: 15 operands: 16
14	ExprTuple	17
15	Literal
16	ExprTuple	58, 18
17	Lambda	parameters: 19 body: 20
18	Operation	operator: 21 operand: 46
19	ExprTuple	23
20	Conditional	value: 24 condition: 25
21	Literal
22	ExprTuple	46
23	ExprRange	lambda_map: 26 start_index: 53 end_index: 54
24	Operation	operator: 27 operand: 30
25	Operation	operator: 41 operands: 29
26	Lambda	parameter: 66 body: 61
27	Literal
28	ExprTuple	30
29	ExprTuple	31
30	Lambda	parameters: 32 body: 33
31	ExprRange	lambda_map: 34 start_index: 53 end_index: 54
32	ExprTuple	35
33	Conditional	value: 36 condition: 37
34	Lambda	parameter: 66 body: 38
35	ExprRange	lambda_map: 39 start_index: 53 end_index: 54
36	Operation	operator: 55 operands: 40
37	Operation	operator: 41 operands: 42
38	Operation	operator: 55 operands: 43
39	Lambda	parameter: 66 body: 62
40	ExprTuple	44, 58
41	Literal
42	ExprTuple	45
43	ExprTuple	61, 46
44	Operation	operator: 47 operands: 48
45	ExprRange	lambda_map: 49 start_index: 53 end_index: 54
46	Variable
47	Literal
48	ExprTuple	50
49	Lambda	parameter: 66 body: 51
50	ExprRange	lambda_map: 52 start_index: 53 end_index: 54
51	Operation	operator: 55 operands: 56
52	Lambda	parameter: 66 body: 57
53	Literal
54	Variable
55	Literal
56	ExprTuple	62, 58
57	Operation	operator: 59 operands: 60
58	Variable
59	Literal
60	ExprTuple	61, 62
61	IndexedVar	variable: 63 index: 66
62	IndexedVar	variable: 64 index: 66
63	Variable
64	Variable
65	ExprTuple	66
66	Variable