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, K, 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, InSet
from proveit.numbers import Natural

# build up the expression from sub-expressions
sub_expr1 = [U_1_to_i, V_1_to_j, W_1_to_k]
expr = Conditional(Forall(instance_param_or_params = sub_expr1, instance_expr = 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 = sub_expr1).with_wrapping(), domain = VecSpaces(K)).with_wrapping(), And(InSet(i, Natural), InSet(j, Natural), InSet(k, Natural)))

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_{U_{1}, U_{2}, \ldots, U_{i}, V_{1}, V_{2}, \ldots, V_{j}, W_{1}, W_{2}, \ldots, W_{k} \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\\
\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}\right]\end{array} \textrm{ if } i \in \mathbb{N} ,  j \in \mathbb{N} ,  k \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: 23 operand: 5
2	Operation	operator: 41 operands: 4
3	ExprTuple	5
4	ExprTuple	6, 7, 8
5	Lambda	parameters: 9 body: 10
6	Operation	operator: 75 operands: 11
7	Operation	operator: 75 operands: 12
8	Operation	operator: 75 operands: 13
9	ExprTuple	14, 15, 16
10	Conditional	value: 17 condition: 18
11	ExprTuple	68, 19
12	ExprTuple	85, 19
13	ExprTuple	72, 19
14	ExprRange	lambda_map: 20 start_index: 84 end_index: 68
15	ExprRange	lambda_map: 21 start_index: 84 end_index: 85
16	ExprRange	lambda_map: 22 start_index: 84 end_index: 72
17	Operation	operator: 23 operand: 26
18	Operation	operator: 41 operands: 25
19	Literal
20	Lambda	parameter: 94 body: 79
21	Lambda	parameter: 94 body: 80
22	Lambda	parameter: 94 body: 82
23	Literal
24	ExprTuple	26
25	ExprTuple	27, 28, 29
26	Lambda	parameters: 53 body: 30
27	ExprRange	lambda_map: 31 start_index: 84 end_index: 68
28	ExprRange	lambda_map: 32 start_index: 84 end_index: 85
29	ExprRange	lambda_map: 33 start_index: 84 end_index: 72
30	Conditional	value: 34 condition: 35
31	Lambda	parameter: 94 body: 36
32	Lambda	parameter: 94 body: 37
33	Lambda	parameter: 94 body: 38
34	Operation	operator: 39 operands: 40
35	Operation	operator: 41 operands: 42
36	Operation	operator: 45 operands: 43
37	Operation	operator: 45 operands: 44
38	Operation	operator: 45 operands: 46
39	Literal
40	ExprTuple	47, 48
41	Literal
42	ExprTuple	49, 50, 51
43	ExprTuple	79, 52
44	ExprTuple	80, 52
45	Literal
46	ExprTuple	82, 52
47	Operation	operator: 69 operands: 53
48	Operation	operator: 69 operands: 54
49	ExprRange	lambda_map: 55 start_index: 84 end_index: 68
50	ExprRange	lambda_map: 56 start_index: 84 end_index: 85
51	ExprRange	lambda_map: 57 start_index: 84 end_index: 72
52	Operation	operator: 58 operand: 66
53	ExprTuple	60, 77, 62
54	ExprTuple	60, 61, 62
55	Lambda	parameter: 94 body: 63
56	Lambda	parameter: 94 body: 64
57	Lambda	parameter: 94 body: 65
58	Literal
59	ExprTuple	66
60	ExprRange	lambda_map: 67 start_index: 84 end_index: 68
61	Operation	operator: 69 operands: 70
62	ExprRange	lambda_map: 71 start_index: 84 end_index: 72
63	Operation	operator: 75 operands: 73
64	Operation	operator: 75 operands: 74
65	Operation	operator: 75 operands: 76
66	Variable
67	Lambda	parameter: 94 body: 78
68	Variable
69	Literal
70	ExprTuple	77
71	Lambda	parameter: 94 body: 81
72	Variable
73	ExprTuple	78, 79
74	ExprTuple	91, 80
75	Literal
76	ExprTuple	81, 82
77	ExprRange	lambda_map: 83 start_index: 84 end_index: 85
78	IndexedVar	variable: 86 index: 94
79	IndexedVar	variable: 87 index: 94
80	IndexedVar	variable: 88 index: 94
81	IndexedVar	variable: 89 index: 94
82	IndexedVar	variable: 90 index: 94
83	Lambda	parameter: 94 body: 91
84	Literal
85	Variable
86	Variable
87	Variable
88	Variable
89	Variable
90	Variable
91	IndexedVar	variable: 92 index: 94
92	Variable
93	ExprTuple	94
94	Variable