Expression of type Forall

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 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, 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 = 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))

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())

\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]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 29 body: 4
3	ExprTuple	29
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 11
6	Operation	operator: 9 operands: 10
7	Literal
8	ExprTuple	11
9	Literal
10	ExprTuple	29, 12
11	Lambda	parameter: 52 body: 14
12	Operation	operator: 15 operand: 24
13	ExprTuple	52
14	Conditional	value: 17 condition: 18
15	Literal
16	ExprTuple	24
17	Operation	operator: 19 operands: 20
18	Operation	operator: 25 operands: 21
19	Literal
20	ExprTuple	22, 23
21	ExprTuple	52, 24
22	Operation	operator: 25 operands: 26
23	Operation	operator: 27 operands: 28
24	Variable
25	Literal
26	ExprTuple	34, 29
27	Literal
28	ExprTuple	30, 31
29	Variable
30	Operation	operator: 46 operands: 32
31	Operation	operator: 36 operand: 35
32	ExprTuple	52, 34
33	ExprTuple	35
34	Operation	operator: 36 operand: 39
35	Lambda	parameters: 56 body: 38
36	Literal
37	ExprTuple	39
38	Conditional	value: 40 condition: 44
39	Lambda	parameters: 56 body: 41
40	Operation	operator: 46 operands: 42
41	Conditional	value: 43 condition: 44
42	ExprTuple	45, 51
43	Operation	operator: 46 operands: 47
44	Operation	operator: 48 operands: 56
45	Operation	operator: 49 operands: 50
46	Literal
47	ExprTuple	53, 51
48	Variable
49	Literal
50	ExprTuple	52, 53
51	Operation	operator: 54 operands: 56
52	Variable
53	Operation	operator: 55 operands: 56
54	Variable
55	Variable
56	ExprTuple	57
57	ExprRange	lambda_map: 58 start_index: 59 end_index: 60
58	Lambda	parameter: 64 body: 61
59	Literal
60	Variable
61	IndexedVar	variable: 62 index: 64
62	Variable
63	ExprTuple	64
64	Variable