Expression of type Forall

from the theory of proveit.linear_algebra.addition

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
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 VecSpaces
from proveit.linear_algebra.addition import vec_summation_b1toj_fQ
from proveit.logic import Forall, Implies, InSet
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [j], instance_expr = Forall(instance_param_or_params = [K, f, Q], instance_expr = Forall(instance_param_or_params = [V], instance_expr = Implies(Forall(instance_param_or_params = [b_1_to_j], instance_expr = InSet(f__b_1_to_j, V), condition = Q__b_1_to_j), InSet(vec_summation_b1toj_fQ, V)).with_wrapping_at(2), 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_{K, f, Q}~\left[\forall_{V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\left(\begin{array}{c} \begin{array}{l} \left[\forall_{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) \in V\right)\right] \Rightarrow  \\ \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) \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: 26 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 51 body: 4
3	ExprTuple	51
4	Conditional	value: 5 condition: 6
5	Operation	operator: 26 operand: 9
6	Operation	operator: 39 operands: 8
7	ExprTuple	9
8	ExprTuple	51, 10
9	Lambda	parameters: 11 body: 12
10	Literal
11	ExprTuple	33, 45, 46
12	Operation	operator: 26 operand: 14
13	ExprTuple	14
14	Lambda	parameter: 42 body: 16
15	ExprTuple	42
16	Conditional	value: 17 condition: 18
17	Operation	operator: 19 operands: 20
18	Operation	operator: 21 operands: 22
19	Literal
20	ExprTuple	23, 24
21	Literal
22	ExprTuple	42, 25
23	Operation	operator: 26 operand: 31
24	Operation	operator: 39 operands: 28
25	Operation	operator: 29 operand: 33
26	Literal
27	ExprTuple	31
28	ExprTuple	32, 42
29	Literal
30	ExprTuple	33
31	Lambda	parameters: 47 body: 34
32	Operation	operator: 35 operand: 38
33	Variable
34	Conditional	value: 37 condition: 44
35	Literal
36	ExprTuple	38
37	Operation	operator: 39 operands: 40
38	Lambda	parameters: 47 body: 41
39	Literal
40	ExprTuple	43, 42
41	Conditional	value: 43 condition: 44
42	Variable
43	Operation	operator: 45 operands: 47
44	Operation	operator: 46 operands: 47
45	Variable
46	Variable
47	ExprTuple	48
48	ExprRange	lambda_map: 49 start_index: 50 end_index: 51
49	Lambda	parameter: 55 body: 52
50	Literal
51	Variable
52	IndexedVar	variable: 53 index: 55
53	Variable
54	ExprTuple	55
55	Variable