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