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, x
from proveit.core_expr_types import x_1_to_n
from proveit.linear_algebra import ScalarMult, VecAdd
from proveit.logic import Equals, Forall, InSet
from proveit.numbers import one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Conditional(Forall(instance_param_or_params = [x_1_to_n], instance_expr = Equals(ScalarMult(a, VecAdd(x_1_to_n)), VecAdd(ExprRange(sub_expr1, ScalarMult(a, IndexedVar(x, sub_expr1)), one, n))).with_wrapping_at(2), domain = V), InSet(a, 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())

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