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, 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.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Mult, NaturalPos

# 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 = [V], instance_expr = Forall(instance_param_or_params = [k], instance_expr = Implies(InSet(sub_expr3, V), Equals(ScalarMult(k, sub_expr3), VecSum(index_or_indices = sub_expr1, summand = ScalarMult(Mult(k, sub_expr2), f__b_1_to_j), condition = Q__b_1_to_j)).with_wrapping_at(1)).with_wrapping_at(1), 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)}~\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(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) \\  = \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] \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	64, 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: 56 body: 18
16	Operation	operator: 19 operand: 28
17	ExprTuple	56
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	56, 28
26	Operation	operator: 29 operands: 30
27	Operation	operator: 31 operands: 32
28	Variable
29	Literal
30	ExprTuple	38, 33
31	Literal
32	ExprTuple	34, 35
33	Variable
34	Operation	operator: 50 operands: 36
35	Operation	operator: 40 operand: 39
36	ExprTuple	56, 38
37	ExprTuple	39
38	Operation	operator: 40 operand: 43
39	Lambda	parameters: 60 body: 42
40	Literal
41	ExprTuple	43
42	Conditional	value: 44 condition: 48
43	Lambda	parameters: 60 body: 45
44	Operation	operator: 50 operands: 46
45	Conditional	value: 47 condition: 48
46	ExprTuple	49, 55
47	Operation	operator: 50 operands: 51
48	Operation	operator: 52 operands: 60
49	Operation	operator: 53 operands: 54
50	Literal
51	ExprTuple	57, 55
52	Variable
53	Literal
54	ExprTuple	56, 57
55	Operation	operator: 58 operands: 60
56	Variable
57	Operation	operator: 59 operands: 60
58	Variable
59	Variable
60	ExprTuple	61
61	ExprRange	lambda_map: 62 start_index: 63 end_index: 64
62	Lambda	parameter: 68 body: 65
63	Literal
64	Variable
65	IndexedVar	variable: 66 index: 68
66	Variable
67	ExprTuple	68
68	Variable