Expression of type Conditional

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 Conditional, 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 = Conditional(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))), InSet(j, 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())

\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] \textrm{ if } j \in \mathbb{N}^+\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 22 operand: 5
2	Operation	operator: 35 operands: 4
3	ExprTuple	5
4	ExprTuple	47, 6
5	Lambda	parameters: 7 body: 8
6	Literal
7	ExprTuple	29, 41, 42
8	Operation	operator: 22 operand: 10
9	ExprTuple	10
10	Lambda	parameter: 38 body: 12
11	ExprTuple	38
12	Conditional	value: 13 condition: 14
13	Operation	operator: 15 operands: 16
14	Operation	operator: 17 operands: 18
15	Literal
16	ExprTuple	19, 20
17	Literal
18	ExprTuple	38, 21
19	Operation	operator: 22 operand: 27
20	Operation	operator: 35 operands: 24
21	Operation	operator: 25 operand: 29
22	Literal
23	ExprTuple	27
24	ExprTuple	28, 38
25	Literal
26	ExprTuple	29
27	Lambda	parameters: 43 body: 30
28	Operation	operator: 31 operand: 34
29	Variable
30	Conditional	value: 33 condition: 40
31	Literal
32	ExprTuple	34
33	Operation	operator: 35 operands: 36
34	Lambda	parameters: 43 body: 37
35	Literal
36	ExprTuple	39, 38
37	Conditional	value: 39 condition: 40
38	Variable
39	Operation	operator: 41 operands: 43
40	Operation	operator: 42 operands: 43
41	Variable
42	Variable
43	ExprTuple	44
44	ExprRange	lambda_map: 45 start_index: 46 end_index: 47
45	Lambda	parameter: 51 body: 48
46	Literal
47	Variable
48	IndexedVar	variable: 49 index: 51
49	Variable
50	ExprTuple	51
51	Variable