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 Conditional, ExprRange, IndexedVar, K, U, V, Variable, W, i, j, k
from proveit.core_expr_types import U_1_to_i, V_1_to_j, W_1_to_k, a_1_to_i, b_1_to_j, c_1_to_k
from proveit.linear_algebra import 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 = VecSpaces(K)
expr = Conditional(Forall(instance_param_or_params = [a_1_to_i, b_1_to_j, c_1_to_k], instance_expr = Equals(TensorProd(a_1_to_i, b_1_to_j, c_1_to_k), TensorProd(a_1_to_i, TensorProd(b_1_to_j), c_1_to_k)).with_wrapping_at(2), domains = [U_1_to_i, V_1_to_j, W_1_to_k]).with_wrapping(), And(ExprRange(sub_expr1, InClass(IndexedVar(U, sub_expr1), sub_expr2), one, i), ExprRange(sub_expr1, InClass(IndexedVar(V, sub_expr1), sub_expr2), one, j), ExprRange(sub_expr1, InClass(IndexedVar(W, sub_expr1), sub_expr2), one, k)))

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