Expression of type Conditional

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, Conditional, 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 And, Equals, Forall, InSet
from proveit.numbers import one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = IndexedVar(A, sub_expr1)
expr = Conditional(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(sub_expr2, [IndexedVar(v, sub_expr1)]), one, n))), domains = [V_1_to_n]).with_wrapping(), And(ExprRange(sub_expr1, InSet(sub_expr2, LinMap(IndexedVar(V, sub_expr1), IndexedVar(W, sub_expr1))), one, n)))

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())

\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} \textrm{ if } \left(A_{1} \in \mathcal{L}\left(V_{1}, W_{1}\right)\right) \land  \left(A_{2} \in \mathcal{L}\left(V_{2}, W_{2}\right)\right) \land  \ldots \land  \left(A_{n} \in \mathcal{L}\left(V_{n}, W_{n}\right)\right)\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operand: 6
2	Operation	operator: 15 operands: 5
3	Literal
4	ExprTuple	6
5	ExprTuple	7
6	Lambda	parameters: 35 body: 8
7	ExprRange	lambda_map: 9 start_index: 45 end_index: 46
8	Conditional	value: 10 condition: 11
9	Lambda	parameter: 54 body: 12
10	Operation	operator: 13 operands: 14
11	Operation	operator: 15 operands: 16
12	Operation	operator: 37 operands: 17
13	Literal
14	ExprTuple	18, 19
15	Literal
16	ExprTuple	20
17	ExprTuple	47, 21
18	Operation	operator: 22 operand: 29
19	Operation	operator: 34 operands: 24
20	ExprRange	lambda_map: 25 start_index: 45 end_index: 46
21	Operation	operator: 26 operands: 27
22	Operation	operator: 34 operands: 28
23	ExprTuple	29
24	ExprTuple	30
25	Lambda	parameter: 54 body: 31
26	Literal
27	ExprTuple	43, 32
28	ExprTuple	33
29	Operation	operator: 34 operands: 35
30	ExprRange	lambda_map: 36 start_index: 45 end_index: 46
31	Operation	operator: 37 operands: 38
32	IndexedVar	variable: 39 index: 54
33	ExprRange	lambda_map: 40 start_index: 45 end_index: 46
34	Literal
35	ExprTuple	41
36	Lambda	parameter: 54 body: 42
37	Literal
38	ExprTuple	51, 43
39	Variable
40	Lambda	parameter: 54 body: 47
41	ExprRange	lambda_map: 44 start_index: 45 end_index: 46
42	Operation	operator: 47 operand: 51
43	IndexedVar	variable: 49 index: 54
44	Lambda	parameter: 54 body: 51
45	Literal
46	Variable
47	IndexedVar	variable: 50 index: 54
48	ExprTuple	51
49	Variable
50	Variable
51	IndexedVar	variable: 52 index: 54
52	Variable
53	ExprTuple	54
54	Variable