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, 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 = [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)

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_{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)

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: 41 body: 5
4	ExprTuple	41
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	41, 13
11	Operation	operator: 14 operands: 15
12	Operation	operator: 16 operands: 17
13	Variable
14	Literal
15	ExprTuple	23, 18
16	Literal
17	ExprTuple	19, 20
18	Variable
19	Operation	operator: 35 operands: 21
20	Operation	operator: 25 operand: 24
21	ExprTuple	41, 23
22	ExprTuple	24
23	Operation	operator: 25 operand: 28
24	Lambda	parameters: 45 body: 27
25	Literal
26	ExprTuple	28
27	Conditional	value: 29 condition: 33
28	Lambda	parameters: 45 body: 30
29	Operation	operator: 35 operands: 31
30	Conditional	value: 32 condition: 33
31	ExprTuple	34, 40
32	Operation	operator: 35 operands: 36
33	Operation	operator: 37 operands: 45
34	Operation	operator: 38 operands: 39
35	Literal
36	ExprTuple	42, 40
37	Variable
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