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, i
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
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
sub_expr1 = [V_1_to_i]
expr = Forall(instance_param_or_params = [i], instance_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)), 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_{i \in \mathbb{N}^+}~\left[\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]\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: 14 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 46 body: 4
3	ExprTuple	46
4	Conditional	value: 5 condition: 6
5	Operation	operator: 14 operand: 9
6	Operation	operator: 47 operands: 8
7	ExprTuple	9
8	ExprTuple	46, 10
9	Lambda	parameters: 35 body: 11
10	Literal
11	Conditional	value: 12 condition: 13
12	Operation	operator: 14 operand: 17
13	Operation	operator: 25 operands: 16
14	Literal
15	ExprTuple	17
16	ExprTuple	18
17	Lambda	parameters: 33 body: 19
18	ExprRange	lambda_map: 20 start_index: 45 end_index: 46
19	Conditional	value: 21 condition: 22
20	Lambda	parameter: 54 body: 23
21	Operation	operator: 47 operands: 24
22	Operation	operator: 25 operands: 26
23	Operation	operator: 27 operands: 28
24	ExprTuple	29, 30
25	Literal
26	ExprTuple	31
27	Literal
28	ExprTuple	50, 32
29	Operation	operator: 34 operands: 33
30	Operation	operator: 34 operands: 35
31	ExprRange	lambda_map: 36 start_index: 45 end_index: 46
32	Operation	operator: 37 operand: 42
33	ExprTuple	39
34	Literal
35	ExprTuple	40
36	Lambda	parameter: 54 body: 41
37	Literal
38	ExprTuple	42
39	ExprRange	lambda_map: 43 start_index: 45 end_index: 46
40	ExprRange	lambda_map: 44 start_index: 45 end_index: 46
41	Operation	operator: 47 operands: 48
42	Variable
43	Lambda	parameter: 54 body: 49
44	Lambda	parameter: 54 body: 50
45	Literal
46	Variable
47	Literal
48	ExprTuple	49, 50
49	IndexedVar	variable: 51 index: 54
50	IndexedVar	variable: 52 index: 54
51	Variable
52	Variable
53	ExprTuple	54
54	Variable