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, ExprRange, IndexedVar, K, V, Variable, a, n, v
from proveit.core_expr_types import a_1_to_n
from proveit.linear_algebra import ScalarMult, VecAdd
from proveit.logic import Equals, Forall, InSet
from proveit.numbers import Natural, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Conditional(Forall(instance_param_or_params = [a_1_to_n], instance_expr = Forall(instance_param_or_params = [v], instance_expr = Equals(ScalarMult(VecAdd(a_1_to_n), v), VecAdd(ExprRange(sub_expr1, ScalarMult(IndexedVar(a, sub_expr1), v), one, n))).with_wrapping_at(2), domain = V), domain = K), InSet(n, Natural))

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_{a_{1}, a_{2}, \ldots, a_{n} \in K}~\left[\forall_{v \in V}~\left(\begin{array}{c} \begin{array}{l} \left(\left(a_{1} +  a_{2} +  \ldots +  a_{n}\right) \cdot v\right) =  \\ \left(\left(a_{1} \cdot v\right) +  \left(a_{2} \cdot v\right) +  \ldots +  \left(a_{n} \cdot v\right)\right) \end{array} \end{array}\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: 10 operand: 5
2	Operation	operator: 25 operands: 4
3	ExprTuple	5
4	ExprTuple	42, 6
5	Lambda	parameters: 36 body: 7
6	Literal
7	Conditional	value: 8 condition: 9
8	Operation	operator: 10 operand: 14
9	Operation	operator: 12 operands: 13
10	Literal
11	ExprTuple	14
12	Literal
13	ExprTuple	15
14	Lambda	parameter: 46 body: 17
15	ExprRange	lambda_map: 18 start_index: 41 end_index: 42
16	ExprTuple	46
17	Conditional	value: 19 condition: 20
18	Lambda	parameter: 49 body: 21
19	Operation	operator: 22 operands: 23
20	Operation	operator: 25 operands: 24
21	Operation	operator: 25 operands: 26
22	Literal
23	ExprTuple	27, 28
24	ExprTuple	46, 29
25	Literal
26	ExprTuple	45, 30
27	Operation	operator: 43 operands: 31
28	Operation	operator: 35 operands: 32
29	Variable
30	Variable
31	ExprTuple	33, 46
32	ExprTuple	34
33	Operation	operator: 35 operands: 36
34	ExprRange	lambda_map: 37 start_index: 41 end_index: 42
35	Literal
36	ExprTuple	38
37	Lambda	parameter: 49 body: 39
38	ExprRange	lambda_map: 40 start_index: 41 end_index: 42
39	Operation	operator: 43 operands: 44
40	Lambda	parameter: 49 body: 45
41	Literal
42	Variable
43	Literal
44	ExprTuple	45, 46
45	IndexedVar	variable: 47 index: 49
46	Variable
47	Variable
48	ExprTuple	49
49	Variable