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, v
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 Complex, 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 = [a_1_to_i, v], instance_expr = Equals(VecAdd(ExprRange(sub_expr1, ScalarMult(IndexedVar(a, sub_expr1), v), one, i)), ScalarMult(VecAdd(a_1_to_i), v)).with_wrapping_at(2), domains = [Complex, 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_{\left(a_{1} \in \mathbb{C}\right), \left(a_{2} \in \mathbb{C}\right), \ldots, \left(a_{i} \in \mathbb{C}\right), v \in V}~\left(\begin{array}{c} \begin{array}{l} \left(\left(a_{1} \cdot v\right) +  \left(a_{2} \cdot v\right) +  \ldots +  \left(a_{i} \cdot v\right)\right) =  \\ \left(\left(a_{1} +  a_{2} +  \ldots +  a_{i}\right) \cdot v\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: 15 operand: 6
2	Operation	operator: 4 operands: 5
3	ExprTuple	6
4	Literal
5	ExprTuple	39, 7
6	Lambda	parameter: 52 body: 9
7	Operation	operator: 10 operand: 14
8	ExprTuple	52
9	Conditional	value: 12 condition: 13
10	Literal
11	ExprTuple	14
12	Operation	operator: 15 operand: 18
13	Operation	operator: 43 operands: 17
14	Variable
15	Literal
16	ExprTuple	18
17	ExprTuple	52, 19
18	Lambda	parameters: 20 body: 21
19	Literal
20	ExprTuple	46, 53
21	Conditional	value: 22 condition: 23
22	Operation	operator: 24 operands: 25
23	Operation	operator: 26 operands: 27
24	Literal
25	ExprTuple	28, 29
26	Literal
27	ExprTuple	30, 31
28	Operation	operator: 41 operands: 32
29	Operation	operator: 48 operands: 33
30	ExprRange	lambda_map: 34 start_index: 51 end_index: 52
31	Operation	operator: 43 operands: 35
32	ExprTuple	36
33	ExprTuple	37, 53
34	Lambda	parameter: 57 body: 38
35	ExprTuple	53, 39
36	ExprRange	lambda_map: 40 start_index: 51 end_index: 52
37	Operation	operator: 41 operands: 42
38	Operation	operator: 43 operands: 44
39	Variable
40	Lambda	parameter: 57 body: 45
41	Literal
42	ExprTuple	46
43	Literal
44	ExprTuple	54, 47
45	Operation	operator: 48 operands: 49
46	ExprRange	lambda_map: 50 start_index: 51 end_index: 52
47	Literal
48	Literal
49	ExprTuple	54, 53
50	Lambda	parameter: 57 body: 54
51	Literal
52	Variable
53	Variable
54	IndexedVar	variable: 55 index: 57
55	Variable
56	ExprTuple	57
57	Variable