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, Function, K, V, c, k
from proveit.core_expr_types import Q__b_1_to_j, b_1_to_j, f__b_1_to_j
from proveit.linear_algebra import ScalarMult, VecSpaces, VecSum
from proveit.logic import Equals, Forall, Implies, InClass, InSet
from proveit.numbers import Mult

# build up the expression from sub-expressions
sub_expr1 = [b_1_to_j]
sub_expr2 = Function(c, sub_expr1)
sub_expr3 = VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr2, f__b_1_to_j), condition = Q__b_1_to_j)
expr = Conditional(Forall(instance_param_or_params = [k], instance_expr = Implies(InSet(sub_expr3, V), Equals(VecSum(index_or_indices = sub_expr1, summand = ScalarMult(Mult(k, sub_expr2), f__b_1_to_j), condition = Q__b_1_to_j), ScalarMult(k, sub_expr3)).with_wrapping_at(1)).with_wrapping_at(1), domain = K), 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_{k \in K}~\left(\begin{array}{c} \begin{array}{l} \left(\left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(c\left(b_{1}, b_{2}, \ldots, b_{j}\right) \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right)\right] \in V\right) \\  \Rightarrow \left(\begin{array}{c} \begin{array}{l} \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(\left(k \cdot c\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right) \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right)\right] \\  = \left(k \cdot \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(c\left(b_{1}, b_{2}, \ldots, b_{j}\right) \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right)\right]\right) \end{array} \end{array}\right) \end{array} \end{array}\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: 3 operand: 7
2	Operation	operator: 5 operands: 6
3	Literal
4	ExprTuple	7
5	Literal
6	ExprTuple	25, 8
7	Lambda	parameter: 47 body: 10
8	Operation	operator: 11 operand: 20
9	ExprTuple	47
10	Conditional	value: 13 condition: 14
11	Literal
12	ExprTuple	20
13	Operation	operator: 15 operands: 16
14	Operation	operator: 21 operands: 17
15	Literal
16	ExprTuple	18, 19
17	ExprTuple	47, 20
18	Operation	operator: 21 operands: 22
19	Operation	operator: 23 operands: 24
20	Variable
21	Literal
22	ExprTuple	31, 25
23	Literal
24	ExprTuple	26, 27
25	Variable
26	Operation	operator: 33 operand: 30
27	Operation	operator: 44 operands: 29
28	ExprTuple	30
29	ExprTuple	47, 31
30	Lambda	parameters: 52 body: 32
31	Operation	operator: 33 operand: 36
32	Conditional	value: 35 condition: 41
33	Literal
34	ExprTuple	36
35	Operation	operator: 44 operands: 37
36	Lambda	parameters: 52 body: 38
37	ExprTuple	39, 49
38	Conditional	value: 40 condition: 41
39	Operation	operator: 42 operands: 43
40	Operation	operator: 44 operands: 45
41	Operation	operator: 46 operands: 52
42	Literal
43	ExprTuple	47, 48
44	Literal
45	ExprTuple	48, 49
46	Variable
47	Variable
48	Operation	operator: 50 operands: 52
49	Operation	operator: 51 operands: 52
50	Variable
51	Variable
52	ExprTuple	53
53	ExprRange	lambda_map: 54 start_index: 55 end_index: 56
54	Lambda	parameter: 60 body: 57
55	Literal
56	Variable
57	IndexedVar	variable: 58 index: 60
58	Variable
59	ExprTuple	60
60	Variable