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, i, k
from proveit.core_expr_types import a_1_to_i
from proveit.linear_algebra import ScalarMult, VecAdd, VecSpaces
from proveit.logic import Equals, Forall, InClass
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 = [i], instance_expr = Forall(instance_param_or_params = [k, a_1_to_i], instance_expr = Equals(VecAdd(ExprRange(sub_expr1, ScalarMult(k, IndexedVar(a, sub_expr1)), one, i)), ScalarMult(k, VecAdd(a_1_to_i))).with_wrapping_at(2), domains = [K, V]), domain = Natural), InClass(V, 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())

\left\{\forall_{i \in \mathbb{N}}~\left[\forall_{k \in K,\left(a_{1} \in V\right), \left(a_{2} \in V\right), \ldots, \left(a_{i} \in V\right)}~\left(\begin{array}{c} \begin{array}{l} \left(\left(k \cdot a_{1}\right) +  \left(k \cdot a_{2}\right) +  \ldots +  \left(k \cdot a_{i}\right)\right) =  \\ \left(k \cdot \left(a_{1} +  a_{2} +  \ldots +  a_{i}\right)\right) \end{array} \end{array}\right)\right] \textrm{ if } V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)\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: 14 operand: 6
2	Operation	operator: 4 operands: 5
3	ExprTuple	6
4	Literal
5	ExprTuple	46, 7
6	Lambda	parameter: 51 body: 9
7	Operation	operator: 10 operand: 37
8	ExprTuple	51
9	Conditional	value: 12 condition: 13
10	Literal
11	ExprTuple	37
12	Operation	operator: 14 operand: 17
13	Operation	operator: 42 operands: 16
14	Literal
15	ExprTuple	17
16	ExprTuple	51, 18
17	Lambda	parameters: 19 body: 20
18	Literal
19	ExprTuple	52, 45
20	Conditional	value: 21 condition: 22
21	Operation	operator: 23 operands: 24
22	Operation	operator: 25 operands: 26
23	Literal
24	ExprTuple	27, 28
25	Literal
26	ExprTuple	29, 30
27	Operation	operator: 40 operands: 31
28	Operation	operator: 47 operands: 32
29	Operation	operator: 42 operands: 33
30	ExprRange	lambda_map: 34 start_index: 50 end_index: 51
31	ExprTuple	35
32	ExprTuple	52, 36
33	ExprTuple	52, 37
34	Lambda	parameter: 56 body: 38
35	ExprRange	lambda_map: 39 start_index: 50 end_index: 51
36	Operation	operator: 40 operands: 41
37	Variable
38	Operation	operator: 42 operands: 43
39	Lambda	parameter: 56 body: 44
40	Literal
41	ExprTuple	45
42	Literal
43	ExprTuple	53, 46
44	Operation	operator: 47 operands: 48
45	ExprRange	lambda_map: 49 start_index: 50 end_index: 51
46	Variable
47	Literal
48	ExprTuple	52, 53
49	Lambda	parameter: 56 body: 53
50	Literal
51	Variable
52	Variable
53	IndexedVar	variable: 54 index: 56
54	Variable
55	ExprTuple	56
56	Variable