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, K, V, 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.linear_algebra.addition import vec_summation_b1toj_fQ
from proveit.logic import Equals, Forall, Implies, InClass, InSet

# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [k], instance_expr = Implies(InSet(vec_summation_b1toj_fQ, V), Equals(ScalarMult(k, vec_summation_b1toj_fQ), VecSum(index_or_indices = [b_1_to_j], summand = ScalarMult(k, f__b_1_to_j), condition = Q__b_1_to_j)).with_wrapping_at(1)).with_wrapping_at(2), 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)}~f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right] \in V\right) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left(k \cdot \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right]\right) \\  = \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(k \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\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: 41 body: 10
8	Operation	operator: 11 operand: 20
9	ExprTuple	41
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	41, 20
18	Operation	operator: 21 operands: 22
19	Operation	operator: 23 operands: 24
20	Variable
21	Literal
22	ExprTuple	30, 25
23	Literal
24	ExprTuple	26, 27
25	Variable
26	Operation	operator: 38 operands: 28
27	Operation	operator: 32 operand: 31
28	ExprTuple	41, 30
29	ExprTuple	31
30	Operation	operator: 32 operand: 35
31	Lambda	parameters: 45 body: 34
32	Literal
33	ExprTuple	35
34	Conditional	value: 36 condition: 40
35	Lambda	parameters: 45 body: 37
36	Operation	operator: 38 operands: 39
37	Conditional	value: 42 condition: 40
38	Literal
39	ExprTuple	41, 42
40	Operation	operator: 43 operands: 45
41	Variable
42	Operation	operator: 44 operands: 45
43	Variable
44	Variable
45	ExprTuple	46
46	ExprRange	lambda_map: 47 start_index: 48 end_index: 49
47	Lambda	parameter: 53 body: 50
48	Literal
49	Variable
50	IndexedVar	variable: 51 index: 53
51	Variable
52	ExprTuple	53
53	Variable