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
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

# build up the expression from sub-expressions
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)))

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_{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]

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: 18 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	25, 37, 38
4	Operation	operator: 18 operand: 6
5	ExprTuple	6
6	Lambda	parameter: 34 body: 8
7	ExprTuple	34
8	Conditional	value: 9 condition: 10
9	Operation	operator: 11 operands: 12
10	Operation	operator: 13 operands: 14
11	Literal
12	ExprTuple	15, 16
13	Literal
14	ExprTuple	34, 17
15	Operation	operator: 18 operand: 23
16	Operation	operator: 31 operands: 20
17	Operation	operator: 21 operand: 25
18	Literal
19	ExprTuple	23
20	ExprTuple	24, 34
21	Literal
22	ExprTuple	25
23	Lambda	parameters: 39 body: 26
24	Operation	operator: 27 operand: 30
25	Variable
26	Conditional	value: 29 condition: 36
27	Literal
28	ExprTuple	30
29	Operation	operator: 31 operands: 32
30	Lambda	parameters: 39 body: 33
31	Literal
32	ExprTuple	35, 34
33	Conditional	value: 35 condition: 36
34	Variable
35	Operation	operator: 37 operands: 39
36	Operation	operator: 38 operands: 39
37	Variable
38	Variable
39	ExprTuple	40
40	ExprRange	lambda_map: 41 start_index: 42 end_index: 43
41	Lambda	parameter: 47 body: 44
42	Literal
43	Variable
44	IndexedVar	variable: 45 index: 47
45	Variable
46	ExprTuple	47
47	Variable