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 K, V, v
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, VecSum
from proveit.linear_algebra.addition import vec_summation_b1toj_fQ
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Sum

# build up the expression from sub-expressions
sub_expr1 = [b_1_to_j]
expr = Forall(instance_param_or_params = [v], instance_expr = Implies(InSet(vec_summation_b1toj_fQ, K), Equals(ScalarMult(Sum(index_or_indices = sub_expr1, summand = f__b_1_to_j, condition = Q__b_1_to_j), v), VecSum(index_or_indices = sub_expr1, summand = ScalarMult(f__b_1_to_j, v), condition = Q__b_1_to_j)).with_wrapping_at(1)).with_wrapping_at(2), domain = V)

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 \in V}~\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 K\right) \Rightarrow  \\ \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] \cdot v\right) \\  = \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(f\left(b_{1}, b_{2}, \ldots, b_{j}\right) \cdot v\right)\right] \end{array} \end{array}\right) \end{array} \end{array}\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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameter: 37 body: 5
4	ExprTuple	37
5	Conditional	value: 6 condition: 7
6	Operation	operator: 8 operands: 9
7	Operation	operator: 14 operands: 10
8	Literal
9	ExprTuple	11, 12
10	ExprTuple	37, 13
11	Operation	operator: 14 operands: 15
12	Operation	operator: 16 operands: 17
13	Variable
14	Literal
15	ExprTuple	18, 19
16	Literal
17	ExprTuple	20, 21
18	Operation	operator: 23 operand: 30
19	Variable
20	Operation	operator: 33 operands: 22
21	Operation	operator: 23 operand: 26
22	ExprTuple	25, 37
23	Literal
24	ExprTuple	26
25	Operation	operator: 27 operand: 30
26	Lambda	parameters: 40 body: 29
27	Literal
28	ExprTuple	30
29	Conditional	value: 31 condition: 35
30	Lambda	parameters: 40 body: 32
31	Operation	operator: 33 operands: 34
32	Conditional	value: 36 condition: 35
33	Literal
34	ExprTuple	36, 37
35	Operation	operator: 38 operands: 40
36	Operation	operator: 39 operands: 40
37	Variable
38	Variable
39	Variable
40	ExprTuple	41
41	ExprRange	lambda_map: 42 start_index: 43 end_index: 44
42	Lambda	parameter: 48 body: 45
43	Literal
44	Variable
45	IndexedVar	variable: 46 index: 48
46	Variable
47	ExprTuple	48
48	Variable