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, ExprRange, Function, IndexedVar, V, Variable, W, n, v
from proveit.core_expr_types import A_1_to_n, V_1_to_n, v_1_to_n
from proveit.linear_algebra import LinMap, TensorProd
from proveit.logic import Equals, Forall
from proveit.numbers import one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Forall(instance_param_or_params = [A_1_to_n], instance_expr = Forall(instance_param_or_params = [v_1_to_n], instance_expr = Equals(Function(TensorProd(A_1_to_n), [TensorProd(v_1_to_n)]), TensorProd(ExprRange(sub_expr1, Function(IndexedVar(A, sub_expr1), [IndexedVar(v, sub_expr1)]), one, n))), domains = [V_1_to_n]).with_wrapping(), domains = [ExprRange(sub_expr1, LinMap(IndexedVar(V, sub_expr1), IndexedVar(W, sub_expr1)), one, n)]).with_wrapping()

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())

\begin{array}{l}\forall_{\left(A_{1} \in \mathcal{L}\left(V_{1}, W_{1}\right)\right), \left(A_{2} \in \mathcal{L}\left(V_{2}, W_{2}\right)\right), \ldots, \left(A_{n} \in \mathcal{L}\left(V_{n}, W_{n}\right)\right)}~\\
\left[\begin{array}{l}\forall_{\left(v_{1} \in V_{1}\right), \left(v_{2} \in V_{2}\right), \ldots, \left(v_{n} \in V_{n}\right)}~\\
\left(\left(A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{n}\right)\left(v_{1} {\otimes}  v_{2} {\otimes}  \ldots {\otimes}  v_{n}\right) = \left(A_{1}\left(v_{1}\right) {\otimes}  A_{2}\left(v_{2}\right) {\otimes}  \ldots {\otimes}  A_{n}\left(v_{n}\right)\right)\right)\end{array}\right]\end{array}

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	True	('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: 31 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 6 operand: 9
5	Operation	operator: 18 operands: 8
6	Literal
7	ExprTuple	9
8	ExprTuple	10
9	Lambda	parameters: 38 body: 11
10	ExprRange	lambda_map: 12 start_index: 48 end_index: 49
11	Conditional	value: 13 condition: 14
12	Lambda	parameter: 57 body: 15
13	Operation	operator: 16 operands: 17
14	Operation	operator: 18 operands: 19
15	Operation	operator: 40 operands: 20
16	Literal
17	ExprTuple	21, 22
18	Literal
19	ExprTuple	23
20	ExprTuple	50, 24
21	Operation	operator: 25 operand: 32
22	Operation	operator: 37 operands: 27
23	ExprRange	lambda_map: 28 start_index: 48 end_index: 49
24	Operation	operator: 29 operands: 30
25	Operation	operator: 37 operands: 31
26	ExprTuple	32
27	ExprTuple	33
28	Lambda	parameter: 57 body: 34
29	Literal
30	ExprTuple	46, 35
31	ExprTuple	36
32	Operation	operator: 37 operands: 38
33	ExprRange	lambda_map: 39 start_index: 48 end_index: 49
34	Operation	operator: 40 operands: 41
35	IndexedVar	variable: 42 index: 57
36	ExprRange	lambda_map: 43 start_index: 48 end_index: 49
37	Literal
38	ExprTuple	44
39	Lambda	parameter: 57 body: 45
40	Literal
41	ExprTuple	54, 46
42	Variable
43	Lambda	parameter: 57 body: 50
44	ExprRange	lambda_map: 47 start_index: 48 end_index: 49
45	Operation	operator: 50 operand: 54
46	IndexedVar	variable: 52 index: 57
47	Lambda	parameter: 57 body: 54
48	Literal
49	Variable
50	IndexedVar	variable: 53 index: 57
51	ExprTuple	54
52	Variable
53	Variable
54	IndexedVar	variable: 55 index: 57
55	Variable
56	ExprTuple	57
57	Variable