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 = [K], instance_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_{K}~\left[\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]\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: 17 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 45 body: 3
3	Operation	operator: 17 operand: 5
4	ExprTuple	5
5	Lambda	parameter: 49 body: 7
6	ExprTuple	49
7	Conditional	value: 8 condition: 9
8	Operation	operator: 17 operand: 12
9	Operation	operator: 50 operands: 11
10	ExprTuple	12
11	ExprTuple	49, 13
12	Lambda	parameters: 38 body: 14
13	Literal
14	Conditional	value: 15 condition: 16
15	Operation	operator: 17 operand: 20
16	Operation	operator: 28 operands: 19
17	Literal
18	ExprTuple	20
19	ExprTuple	21
20	Lambda	parameters: 36 body: 22
21	ExprRange	lambda_map: 23 start_index: 48 end_index: 49
22	Conditional	value: 24 condition: 25
23	Lambda	parameter: 57 body: 26
24	Operation	operator: 50 operands: 27
25	Operation	operator: 28 operands: 29
26	Operation	operator: 30 operands: 31
27	ExprTuple	32, 33
28	Literal
29	ExprTuple	34
30	Literal
31	ExprTuple	53, 35
32	Operation	operator: 37 operands: 36
33	Operation	operator: 37 operands: 38
34	ExprRange	lambda_map: 39 start_index: 48 end_index: 49
35	Operation	operator: 40 operand: 45
36	ExprTuple	42
37	Literal
38	ExprTuple	43
39	Lambda	parameter: 57 body: 44
40	Literal
41	ExprTuple	45
42	ExprRange	lambda_map: 46 start_index: 48 end_index: 49
43	ExprRange	lambda_map: 47 start_index: 48 end_index: 49
44	Operation	operator: 50 operands: 51
45	Variable
46	Lambda	parameter: 57 body: 52
47	Lambda	parameter: 57 body: 53
48	Literal
49	Variable
50	Literal
51	ExprTuple	52, 53
52	IndexedVar	variable: 54 index: 57
53	IndexedVar	variable: 55 index: 57
54	Variable
55	Variable
56	ExprTuple	57
57	Variable