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, W_1_to_n, v_1_to_n
from proveit.linear_algebra import LinMap, TensorProd, VecSpaces
from proveit.logic import Equals, Forall, InSet
from proveit.numbers import NaturalPos, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Conditional(Forall(instance_param_or_params = [V_1_to_n, W_1_to_n], instance_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(), domain = VecSpaces(K)).with_wrapping(), InSet(n, NaturalPos))

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_{V_{1}, V_{2}, \ldots, V_{n}, W_{1}, W_{2}, \ldots, W_{n} \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\\
\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}\right]\end{array} \textrm{ if } n \in \mathbb{N}^+\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: 27 operand: 5
2	Operation	operator: 68 operands: 4
3	ExprTuple	5
4	ExprTuple	77, 6
5	Lambda	parameters: 7 body: 8
6	Literal
7	ExprTuple	9, 10
8	Conditional	value: 11 condition: 12
9	ExprRange	lambda_map: 13 start_index: 76 end_index: 77
10	ExprRange	lambda_map: 14 start_index: 76 end_index: 77
11	Operation	operator: 27 operand: 17
12	Operation	operator: 46 operands: 16
13	Lambda	parameter: 85 body: 74
14	Lambda	parameter: 85 body: 63
15	ExprTuple	17
16	ExprTuple	18, 19
17	Lambda	parameters: 59 body: 20
18	ExprRange	lambda_map: 21 start_index: 76 end_index: 77
19	ExprRange	lambda_map: 22 start_index: 76 end_index: 77
20	Conditional	value: 23 condition: 24
21	Lambda	parameter: 85 body: 25
22	Lambda	parameter: 85 body: 26
23	Operation	operator: 27 operand: 33
24	Operation	operator: 46 operands: 29
25	Operation	operator: 31 operands: 30
26	Operation	operator: 31 operands: 32
27	Literal
28	ExprTuple	33
29	ExprTuple	34
30	ExprTuple	74, 35
31	Literal
32	ExprTuple	63, 35
33	Lambda	parameters: 66 body: 36
34	ExprRange	lambda_map: 37 start_index: 76 end_index: 77
35	Operation	operator: 38 operand: 43
36	Conditional	value: 40 condition: 41
37	Lambda	parameter: 85 body: 42
38	Literal
39	ExprTuple	43
40	Operation	operator: 44 operands: 45
41	Operation	operator: 46 operands: 47
42	Operation	operator: 68 operands: 48
43	Variable
44	Literal
45	ExprTuple	49, 50
46	Literal
47	ExprTuple	51
48	ExprTuple	78, 52
49	Operation	operator: 53 operand: 60
50	Operation	operator: 65 operands: 55
51	ExprRange	lambda_map: 56 start_index: 76 end_index: 77
52	Operation	operator: 57 operands: 58
53	Operation	operator: 65 operands: 59
54	ExprTuple	60
55	ExprTuple	61
56	Lambda	parameter: 85 body: 62
57	Literal
58	ExprTuple	74, 63
59	ExprTuple	64
60	Operation	operator: 65 operands: 66
61	ExprRange	lambda_map: 67 start_index: 76 end_index: 77
62	Operation	operator: 68 operands: 69
63	IndexedVar	variable: 70 index: 85
64	ExprRange	lambda_map: 71 start_index: 76 end_index: 77
65	Literal
66	ExprTuple	72
67	Lambda	parameter: 85 body: 73
68	Literal
69	ExprTuple	82, 74
70	Variable
71	Lambda	parameter: 85 body: 78
72	ExprRange	lambda_map: 75 start_index: 76 end_index: 77
73	Operation	operator: 78 operand: 82
74	IndexedVar	variable: 80 index: 85
75	Lambda	parameter: 85 body: 82
76	Literal
77	Variable
78	IndexedVar	variable: 81 index: 85
79	ExprTuple	82
80	Variable
81	Variable
82	IndexedVar	variable: 83 index: 85
83	Variable
84	ExprTuple	85
85	Variable