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, V, alpha, b, i, k
from proveit.core_expr_types import U_1_to_i, W_1_to_k, a_1_to_i, c_1_to_k
from proveit.linear_algebra import ScalarMult, 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, 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, c_1_to_k], instance_expr = Equals(TensorProd(a_1_to_i, ScalarMult(alpha, b), c_1_to_k), ScalarMult(alpha, TensorProd(a_1_to_i, b, c_1_to_k))).with_wrapping_at(1), domains = sub_expr1).with_wrapping(), domain = VecSpaces(K)).with_wrapping(), And(InSet(i, 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, 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), b \in V,\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} \left(\alpha \cdot b\right){\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \\  = \left(\alpha \cdot \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} b{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right)\right) \end{array} \end{array}\right)\end{array}\right]\end{array} \textrm{ if } i \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: 19 operand: 5
2	Operation	operator: 36 operands: 4
3	ExprTuple	5
4	ExprTuple	6, 7
5	Lambda	parameters: 8 body: 9
6	Operation	operator: 65 operands: 10
7	Operation	operator: 65 operands: 11
8	ExprTuple	12, 57, 13
9	Conditional	value: 14 condition: 15
10	ExprTuple	74, 16
11	ExprTuple	77, 16
12	ExprRange	lambda_map: 17 start_index: 76 end_index: 74
13	ExprRange	lambda_map: 18 start_index: 76 end_index: 77
14	Operation	operator: 19 operand: 22
15	Operation	operator: 36 operands: 21
16	Literal
17	Lambda	parameter: 85 body: 71
18	Lambda	parameter: 85 body: 72
19	Literal
20	ExprTuple	22
21	ExprTuple	23, 24, 25
22	Lambda	parameters: 63 body: 26
23	ExprRange	lambda_map: 27 start_index: 76 end_index: 74
24	Operation	operator: 39 operands: 28
25	ExprRange	lambda_map: 29 start_index: 76 end_index: 77
26	Conditional	value: 30 condition: 31
27	Lambda	parameter: 85 body: 32
28	ExprTuple	57, 46
29	Lambda	parameter: 85 body: 33
30	Operation	operator: 34 operands: 35
31	Operation	operator: 36 operands: 37
32	Operation	operator: 39 operands: 38
33	Operation	operator: 39 operands: 40
34	Literal
35	ExprTuple	41, 42
36	Literal
37	ExprTuple	43, 44, 45
38	ExprTuple	71, 46
39	Literal
40	ExprTuple	72, 46
41	Operation	operator: 62 operands: 47
42	Operation	operator: 60 operands: 48
43	ExprRange	lambda_map: 49 start_index: 76 end_index: 74
44	Operation	operator: 65 operands: 50
45	ExprRange	lambda_map: 51 start_index: 76 end_index: 77
46	Operation	operator: 52 operand: 59
47	ExprTuple	68, 54, 70
48	ExprTuple	67, 55
49	Lambda	parameter: 85 body: 56
50	ExprTuple	69, 57
51	Lambda	parameter: 85 body: 58
52	Literal
53	ExprTuple	59
54	Operation	operator: 60 operands: 61
55	Operation	operator: 62 operands: 63
56	Operation	operator: 65 operands: 64
57	Variable
58	Operation	operator: 65 operands: 66
59	Variable
60	Literal
61	ExprTuple	67, 69
62	Literal
63	ExprTuple	68, 69, 70
64	ExprTuple	80, 71
65	Literal
66	ExprTuple	81, 72
67	Variable
68	ExprRange	lambda_map: 73 start_index: 76 end_index: 74
69	Variable
70	ExprRange	lambda_map: 75 start_index: 76 end_index: 77
71	IndexedVar	variable: 78 index: 85
72	IndexedVar	variable: 79 index: 85
73	Lambda	parameter: 85 body: 80
74	Variable
75	Lambda	parameter: 85 body: 81
76	Literal
77	Variable
78	Variable
79	Variable
80	IndexedVar	variable: 82 index: 85
81	IndexedVar	variable: 83 index: 85
82	Variable
83	Variable
84	ExprTuple	85
85	Variable