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, 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, 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, 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 = [U_1_to_i, V, W_1_to_k]).with_wrapping(), And(ExprRange(sub_expr1, InClass(IndexedVar(U, sub_expr1), sub_expr2), one, i), InClass(V, sub_expr2), 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), 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} \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) ,  V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\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: 20 operands: 5
3	Literal
4	ExprTuple	6
5	ExprTuple	7, 8, 9
6	Lambda	parameters: 47 body: 10
7	ExprRange	lambda_map: 11 start_index: 60 end_index: 58
8	Operation	operator: 23 operands: 12
9	ExprRange	lambda_map: 13 start_index: 60 end_index: 61
10	Conditional	value: 14 condition: 15
11	Lambda	parameter: 69 body: 16
12	ExprTuple	41, 30
13	Lambda	parameter: 69 body: 17
14	Operation	operator: 18 operands: 19
15	Operation	operator: 20 operands: 21
16	Operation	operator: 23 operands: 22
17	Operation	operator: 23 operands: 24
18	Literal
19	ExprTuple	25, 26
20	Literal
21	ExprTuple	27, 28, 29
22	ExprTuple	55, 30
23	Literal
24	ExprTuple	56, 30
25	Operation	operator: 46 operands: 31
26	Operation	operator: 44 operands: 32
27	ExprRange	lambda_map: 33 start_index: 60 end_index: 58
28	Operation	operator: 49 operands: 34
29	ExprRange	lambda_map: 35 start_index: 60 end_index: 61
30	Operation	operator: 36 operand: 43
31	ExprTuple	52, 38, 54
32	ExprTuple	51, 39
33	Lambda	parameter: 69 body: 40
34	ExprTuple	53, 41
35	Lambda	parameter: 69 body: 42
36	Literal
37	ExprTuple	43
38	Operation	operator: 44 operands: 45
39	Operation	operator: 46 operands: 47
40	Operation	operator: 49 operands: 48
41	Variable
42	Operation	operator: 49 operands: 50
43	Variable
44	Literal
45	ExprTuple	51, 53
46	Literal
47	ExprTuple	52, 53, 54
48	ExprTuple	64, 55
49	Literal
50	ExprTuple	65, 56
51	Variable
52	ExprRange	lambda_map: 57 start_index: 60 end_index: 58
53	Variable
54	ExprRange	lambda_map: 59 start_index: 60 end_index: 61
55	IndexedVar	variable: 62 index: 69
56	IndexedVar	variable: 63 index: 69
57	Lambda	parameter: 69 body: 64
58	Variable
59	Lambda	parameter: 69 body: 65
60	Literal
61	Variable
62	Variable
63	Variable
64	IndexedVar	variable: 66 index: 69
65	IndexedVar	variable: 67 index: 69
66	Variable
67	Variable
68	ExprTuple	69
69	Variable