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, j, 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, InSet
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [V], instance_expr = Forall(instance_param_or_params = [k], instance_expr = Implies(InSet(vec_summation_b1toj_fQ, V), Equals(VecSum(index_or_indices = [b_1_to_j], summand = ScalarMult(k, f__b_1_to_j), condition = Q__b_1_to_j), ScalarMult(k, vec_summation_b1toj_fQ)).with_wrapping_at(1)).with_wrapping_at(2), domain = K), domain = VecSpaces(K)), InSet(j, NaturalPos))

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_{V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\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[\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] \\  = \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) \end{array} \end{array}\right) \end{array} \end{array}\right)\right] \textrm{ if } j \in \mathbb{N}^+\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: 11 operand: 5
2	Operation	operator: 29 operands: 4
3	ExprTuple	5
4	ExprTuple	57, 6
5	Lambda	parameter: 33 body: 8
6	Literal
7	ExprTuple	33
8	Conditional	value: 9 condition: 10
9	Operation	operator: 11 operand: 15
10	Operation	operator: 13 operands: 14
11	Literal
12	ExprTuple	15
13	Literal
14	ExprTuple	33, 16
15	Lambda	parameter: 48 body: 18
16	Operation	operator: 19 operand: 28
17	ExprTuple	48
18	Conditional	value: 21 condition: 22
19	Literal
20	ExprTuple	28
21	Operation	operator: 23 operands: 24
22	Operation	operator: 29 operands: 25
23	Literal
24	ExprTuple	26, 27
25	ExprTuple	48, 28
26	Operation	operator: 29 operands: 30
27	Operation	operator: 31 operands: 32
28	Variable
29	Literal
30	ExprTuple	39, 33
31	Literal
32	ExprTuple	34, 35
33	Variable
34	Operation	operator: 41 operand: 38
35	Operation	operator: 45 operands: 37
36	ExprTuple	38
37	ExprTuple	48, 39
38	Lambda	parameters: 53 body: 40
39	Operation	operator: 41 operand: 44
40	Conditional	value: 43 condition: 50
41	Literal
42	ExprTuple	44
43	Operation	operator: 45 operands: 46
44	Lambda	parameters: 53 body: 47
45	Literal
46	ExprTuple	48, 49
47	Conditional	value: 49 condition: 50
48	Variable
49	Operation	operator: 51 operands: 53
50	Operation	operator: 52 operands: 53
51	Variable
52	Variable
53	ExprTuple	54
54	ExprRange	lambda_map: 55 start_index: 56 end_index: 57
55	Lambda	parameter: 61 body: 58
56	Literal
57	Variable
58	IndexedVar	variable: 59 index: 61
59	Variable
60	ExprTuple	61
61	Variable