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 A, B, K, U, V, W
from proveit.linear_algebra import TensorProd, VecSpaces
from proveit.logic import And, Exists, Forall, InSet

# build up the expression from sub-expressions
sub_expr1 = VecSpaces(K)
expr = Forall(instance_param_or_params = [W], instance_expr = Forall(instance_param_or_params = [A, B], instance_expr = Exists(instance_param_or_params = [U, V], instance_expr = And(InSet(A, U), InSet(B, V)), domain = sub_expr1), condition = InSet(TensorProd(A, B), W)), domain = sub_expr1)

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_{W \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\left[\forall_{A, B~|~\left(A {\otimes} B\right) \in W}~\left[\exists_{U, V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\left(\left(A \in U\right) \land \left(B \in V\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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 19 body: 4
3	ExprTuple	19
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 37 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	19, 43
10	Lambda	parameters: 23 body: 11
11	Conditional	value: 12 condition: 13
12	Operation	operator: 14 operand: 17
13	Operation	operator: 34 operands: 16
14	Literal
15	ExprTuple	17
16	ExprTuple	18, 19
17	Lambda	parameters: 20 body: 21
18	Operation	operator: 22 operands: 23
19	Variable
20	ExprTuple	41, 42
21	Conditional	value: 24 condition: 25
22	Literal
23	ExprTuple	39, 40
24	Operation	operator: 27 operands: 26
25	Operation	operator: 27 operands: 28
26	ExprTuple	29, 30
27	Literal
28	ExprTuple	31, 32
29	Operation	operator: 34 operands: 33
30	Operation	operator: 34 operands: 35
31	Operation	operator: 37 operands: 36
32	Operation	operator: 37 operands: 38
33	ExprTuple	39, 41
34	Literal
35	ExprTuple	40, 42
36	ExprTuple	41, 43
37	Literal
38	ExprTuple	42, 43
39	Variable
40	Variable
41	Variable
42	Variable
43	Operation	operator: 44 operand: 46
44	Literal
45	ExprTuple	46
46	Variable