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, Q, V, f, 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 = Forall(instance_param_or_params = [K, f, Q], instance_expr = Forall(instance_param_or_params = [j], instance_expr = Forall(instance_param_or_params = [V], instance_expr = Forall(instance_param_or_params = [k], instance_expr = Implies(InSet(vec_summation_b1toj_fQ, V), Equals(ScalarMult(k, vec_summation_b1toj_fQ), VecSum(index_or_indices = [b_1_to_j], summand = ScalarMult(k, f__b_1_to_j), condition = Q__b_1_to_j)).with_wrapping_at(1)).with_wrapping_at(2), domain = K), domain = VecSpaces(K)), domain = 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())

\forall_{K, f, Q}~\left[\forall_{j \in \mathbb{N}^+}~\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(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) \\  = \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] \end{array} \end{array}\right) \end{array} \end{array}\right)\right]\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: 19 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	36, 60, 59
4	Operation	operator: 19 operand: 6
5	ExprTuple	6
6	Lambda	parameter: 65 body: 8
7	ExprTuple	65
8	Conditional	value: 9 condition: 10
9	Operation	operator: 19 operand: 13
10	Operation	operator: 37 operands: 12
11	ExprTuple	13
12	ExprTuple	65, 14
13	Lambda	parameter: 41 body: 16
14	Literal
15	ExprTuple	41
16	Conditional	value: 17 condition: 18
17	Operation	operator: 19 operand: 23
18	Operation	operator: 21 operands: 22
19	Literal
20	ExprTuple	23
21	Literal
22	ExprTuple	41, 24
23	Lambda	parameter: 57 body: 26
24	Operation	operator: 27 operand: 36
25	ExprTuple	57
26	Conditional	value: 29 condition: 30
27	Literal
28	ExprTuple	36
29	Operation	operator: 31 operands: 32
30	Operation	operator: 37 operands: 33
31	Literal
32	ExprTuple	34, 35
33	ExprTuple	57, 36
34	Operation	operator: 37 operands: 38
35	Operation	operator: 39 operands: 40
36	Variable
37	Literal
38	ExprTuple	46, 41
39	Literal
40	ExprTuple	42, 43
41	Variable
42	Operation	operator: 54 operands: 44
43	Operation	operator: 48 operand: 47
44	ExprTuple	57, 46
45	ExprTuple	47
46	Operation	operator: 48 operand: 51
47	Lambda	parameters: 61 body: 50
48	Literal
49	ExprTuple	51
50	Conditional	value: 52 condition: 56
51	Lambda	parameters: 61 body: 53
52	Operation	operator: 54 operands: 55
53	Conditional	value: 58 condition: 56
54	Literal
55	ExprTuple	57, 58
56	Operation	operator: 59 operands: 61
57	Variable
58	Operation	operator: 60 operands: 61
59	Variable
60	Variable
61	ExprTuple	62
62	ExprRange	lambda_map: 63 start_index: 64 end_index: 65
63	Lambda	parameter: 69 body: 66
64	Literal
65	Variable
66	IndexedVar	variable: 67 index: 69
67	Variable
68	ExprTuple	69
69	Variable