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, 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 = 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(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)), 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_{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)}~\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]\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: 15 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 68 body: 4
3	ExprTuple	68
4	Conditional	value: 5 condition: 6
5	Operation	operator: 15 operand: 9
6	Operation	operator: 33 operands: 8
7	ExprTuple	9
8	ExprTuple	68, 10
9	Lambda	parameter: 37 body: 12
10	Literal
11	ExprTuple	37
12	Conditional	value: 13 condition: 14
13	Operation	operator: 15 operand: 19
14	Operation	operator: 17 operands: 18
15	Literal
16	ExprTuple	19
17	Literal
18	ExprTuple	37, 20
19	Lambda	parameter: 60 body: 22
20	Operation	operator: 23 operand: 32
21	ExprTuple	60
22	Conditional	value: 25 condition: 26
23	Literal
24	ExprTuple	32
25	Operation	operator: 27 operands: 28
26	Operation	operator: 33 operands: 29
27	Literal
28	ExprTuple	30, 31
29	ExprTuple	60, 32
30	Operation	operator: 33 operands: 34
31	Operation	operator: 35 operands: 36
32	Variable
33	Literal
34	ExprTuple	42, 37
35	Literal
36	ExprTuple	38, 39
37	Variable
38	Operation	operator: 54 operands: 40
39	Operation	operator: 44 operand: 43
40	ExprTuple	60, 42
41	ExprTuple	43
42	Operation	operator: 44 operand: 47
43	Lambda	parameters: 64 body: 46
44	Literal
45	ExprTuple	47
46	Conditional	value: 48 condition: 52
47	Lambda	parameters: 64 body: 49
48	Operation	operator: 54 operands: 50
49	Conditional	value: 51 condition: 52
50	ExprTuple	53, 59
51	Operation	operator: 54 operands: 55
52	Operation	operator: 56 operands: 64
53	Operation	operator: 57 operands: 58
54	Literal
55	ExprTuple	61, 59
56	Variable
57	Literal
58	ExprTuple	60, 61
59	Operation	operator: 62 operands: 64
60	Variable
61	Operation	operator: 63 operands: 64
62	Variable
63	Variable
64	ExprTuple	65
65	ExprRange	lambda_map: 66 start_index: 67 end_index: 68
66	Lambda	parameter: 72 body: 69
67	Literal
68	Variable
69	IndexedVar	variable: 70 index: 72
70	Variable
71	ExprTuple	72
72	Variable