Expression of type Forall

from the theory of proveit.linear_algebra.tensors

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
from proveit.core_expr_types import V_1_to_i, a_1_to_i
from proveit.linear_algebra import TensorProd, VecSpaces
from proveit.logic import Forall, InSet

# build up the expression from sub-expressions
sub_expr1 = [V_1_to_i]
expr = Forall(instance_param_or_params = sub_expr1, instance_expr = Forall(instance_param_or_params = [a_1_to_i], instance_expr = InSet(TensorProd(a_1_to_i), TensorProd(V_1_to_i)), domains = sub_expr1), 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_{V_{1}, V_{2}, \ldots, V_{i} \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\left[\forall_{\left(a_{1} \in V_{1}\right), \left(a_{2} \in V_{2}\right), \ldots, \left(a_{i} \in V_{i}\right)}~\left(\left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i}\right) \in \left(V_{1} {\otimes}  V_{2} {\otimes}  \ldots {\otimes}  V_{i}\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: 6 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 27 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 6 operand: 9
5	Operation	operator: 17 operands: 8
6	Literal
7	ExprTuple	9
8	ExprTuple	10
9	Lambda	parameters: 25 body: 11
10	ExprRange	lambda_map: 12 start_index: 37 end_index: 38
11	Conditional	value: 13 condition: 14
12	Lambda	parameter: 46 body: 15
13	Operation	operator: 39 operands: 16
14	Operation	operator: 17 operands: 18
15	Operation	operator: 19 operands: 20
16	ExprTuple	21, 22
17	Literal
18	ExprTuple	23
19	Literal
20	ExprTuple	42, 24
21	Operation	operator: 26 operands: 25
22	Operation	operator: 26 operands: 27
23	ExprRange	lambda_map: 28 start_index: 37 end_index: 38
24	Operation	operator: 29 operand: 34
25	ExprTuple	31
26	Literal
27	ExprTuple	32
28	Lambda	parameter: 46 body: 33
29	Literal
30	ExprTuple	34
31	ExprRange	lambda_map: 35 start_index: 37 end_index: 38
32	ExprRange	lambda_map: 36 start_index: 37 end_index: 38
33	Operation	operator: 39 operands: 40
34	Variable
35	Lambda	parameter: 46 body: 41
36	Lambda	parameter: 46 body: 42
37	Literal
38	Variable
39	Literal
40	ExprTuple	41, 42
41	IndexedVar	variable: 43 index: 46
42	IndexedVar	variable: 44 index: 46
43	Variable
44	Variable
45	ExprTuple	46
46	Variable