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, K, 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, VecSpaces
from proveit.logic import And, Equals, Forall, InClass
from proveit.numbers import one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = IndexedVar(V, sub_expr1)
sub_expr3 = IndexedVar(W, sub_expr1)
sub_expr4 = VecSpaces(K)
expr = Conditional(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(sub_expr2, sub_expr3), one, n)]).with_wrapping(), And(ExprRange(sub_expr1, InClass(sub_expr2, sub_expr4), one, n), ExprRange(sub_expr1, InClass(sub_expr3, sub_expr4), 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(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} \textrm{ if } \left(V_{1} \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)\right) ,  \left(V_{2} \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)\right) ,  \ldots ,  \left(V_{n} \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)\right), \left(W_{1} \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)\right) ,  \left(W_{2} \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)\right) ,  \ldots ,  \left(W_{n} \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\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: 15 operand: 5
2	Operation	operator: 34 operands: 4
3	ExprTuple	5
4	ExprTuple	6, 7
5	Lambda	parameters: 47 body: 8
6	ExprRange	lambda_map: 9 start_index: 64 end_index: 65
7	ExprRange	lambda_map: 10 start_index: 64 end_index: 65
8	Conditional	value: 11 condition: 12
9	Lambda	parameter: 73 body: 13
10	Lambda	parameter: 73 body: 14
11	Operation	operator: 15 operand: 21
12	Operation	operator: 34 operands: 17
13	Operation	operator: 19 operands: 18
14	Operation	operator: 19 operands: 20
15	Literal
16	ExprTuple	21
17	ExprTuple	22
18	ExprTuple	62, 23
19	Literal
20	ExprTuple	51, 23
21	Lambda	parameters: 54 body: 24
22	ExprRange	lambda_map: 25 start_index: 64 end_index: 65
23	Operation	operator: 26 operand: 31
24	Conditional	value: 28 condition: 29
25	Lambda	parameter: 73 body: 30
26	Literal
27	ExprTuple	31
28	Operation	operator: 32 operands: 33
29	Operation	operator: 34 operands: 35
30	Operation	operator: 56 operands: 36
31	Variable
32	Literal
33	ExprTuple	37, 38
34	Literal
35	ExprTuple	39
36	ExprTuple	66, 40
37	Operation	operator: 41 operand: 48
38	Operation	operator: 53 operands: 43
39	ExprRange	lambda_map: 44 start_index: 64 end_index: 65
40	Operation	operator: 45 operands: 46
41	Operation	operator: 53 operands: 47
42	ExprTuple	48
43	ExprTuple	49
44	Lambda	parameter: 73 body: 50
45	Literal
46	ExprTuple	62, 51
47	ExprTuple	52
48	Operation	operator: 53 operands: 54
49	ExprRange	lambda_map: 55 start_index: 64 end_index: 65
50	Operation	operator: 56 operands: 57
51	IndexedVar	variable: 58 index: 73
52	ExprRange	lambda_map: 59 start_index: 64 end_index: 65
53	Literal
54	ExprTuple	60
55	Lambda	parameter: 73 body: 61
56	Literal
57	ExprTuple	70, 62
58	Variable
59	Lambda	parameter: 73 body: 66
60	ExprRange	lambda_map: 63 start_index: 64 end_index: 65
61	Operation	operator: 66 operand: 70
62	IndexedVar	variable: 68 index: 73
63	Lambda	parameter: 73 body: 70
64	Literal
65	Variable
66	IndexedVar	variable: 69 index: 73
67	ExprTuple	70
68	Variable
69	Variable
70	IndexedVar	variable: 71 index: 73
71	Variable
72	ExprTuple	73
73	Variable