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, U, V, Variable, W, a, b, c, d, e, i, k
from proveit.core_expr_types import a_1_to_i, c_1_to_k, d_1_to_i, e_1_to_k
from proveit.linear_algebra import TensorProd
from proveit.logic import And, Equals, Forall, Implies, InSet
from proveit.numbers import one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = IndexedVar(U, sub_expr1)
sub_expr3 = IndexedVar(W, sub_expr1)
expr = Conditional(Forall(instance_param_or_params = [b], instance_expr = Implies(Equals(TensorProd(a_1_to_i, c_1_to_k), TensorProd(d_1_to_i, e_1_to_k)).with_wrapping_at(2), Equals(TensorProd(a_1_to_i, b, c_1_to_k), TensorProd(d_1_to_i, b, e_1_to_k)).with_wrapping_at(2)).with_wrapping_at(2), domain = V), And(ExprRange(sub_expr1, InSet(IndexedVar(a, sub_expr1), sub_expr2), one, i), ExprRange(sub_expr1, InSet(IndexedVar(c, sub_expr1), sub_expr3), one, k), ExprRange(sub_expr1, InSet(IndexedVar(d, sub_expr1), sub_expr2), one, i), ExprRange(sub_expr1, InSet(IndexedVar(e, sub_expr1), sub_expr3), 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\{\forall_{b \in V}~\left(\begin{array}{c} \begin{array}{l} \left(\begin{array}{c} \begin{array}{l} \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i}{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) =  \\ \left(d_{1} {\otimes}  d_{2} {\otimes}  \ldots {\otimes}  d_{i}{\otimes} e_{1} {\otimes}  e_{2} {\otimes}  \ldots {\otimes}  e_{k}\right) \end{array} \end{array}\right) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \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) =  \\ \left(d_{1} {\otimes}  d_{2} {\otimes}  \ldots {\otimes}  d_{i} {\otimes} b{\otimes} e_{1} {\otimes}  e_{2} {\otimes}  \ldots {\otimes}  e_{k}\right) \end{array} \end{array}\right) \end{array} \end{array}\right) \textrm{ if } \left(a_{1} \in U_{1}\right) ,  \left(a_{2} \in U_{2}\right) ,  \ldots ,  \left(a_{i} \in U_{i}\right), \left(c_{1} \in W_{1}\right) ,  \left(c_{2} \in W_{2}\right) ,  \ldots ,  \left(c_{k} \in W_{k}\right), \left(d_{1} \in U_{1}\right) ,  \left(d_{2} \in U_{2}\right) ,  \ldots ,  \left(d_{i} \in U_{i}\right), \left(e_{1} \in W_{1}\right) ,  \left(e_{2} \in W_{2}\right) ,  \ldots ,  \left(e_{k} \in W_{k}\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: 7
2	Operation	operator: 5 operands: 6
3	Literal
4	ExprTuple	7
5	Literal
6	ExprTuple	8, 9, 10, 11
7	Lambda	parameter: 54 body: 13
8	ExprRange	lambda_map: 14 start_index: 61 end_index: 59
9	ExprRange	lambda_map: 15 start_index: 61 end_index: 62
10	ExprRange	lambda_map: 16 start_index: 61 end_index: 59
11	ExprRange	lambda_map: 17 start_index: 61 end_index: 62
12	ExprTuple	54
13	Conditional	value: 18 condition: 19
14	Lambda	parameter: 72 body: 20
15	Lambda	parameter: 72 body: 21
16	Lambda	parameter: 72 body: 22
17	Lambda	parameter: 72 body: 23
18	Operation	operator: 24 operands: 25
19	Operation	operator: 30 operands: 26
20	Operation	operator: 30 operands: 27
21	Operation	operator: 30 operands: 28
22	Operation	operator: 30 operands: 29
23	Operation	operator: 30 operands: 31
24	Literal
25	ExprTuple	32, 33
26	ExprTuple	54, 34
27	ExprTuple	63, 35
28	ExprTuple	64, 36
29	ExprTuple	65, 35
30	Literal
31	ExprTuple	66, 36
32	Operation	operator: 38 operands: 37
33	Operation	operator: 38 operands: 39
34	Variable
35	IndexedVar	variable: 40 index: 72
36	IndexedVar	variable: 41 index: 72
37	ExprTuple	42, 43
38	Literal
39	ExprTuple	44, 45
40	Variable
41	Variable
42	Operation	operator: 49 operands: 46
43	Operation	operator: 49 operands: 47
44	Operation	operator: 49 operands: 48
45	Operation	operator: 49 operands: 50
46	ExprTuple	51, 52
47	ExprTuple	53, 55
48	ExprTuple	51, 54, 52
49	Literal
50	ExprTuple	53, 54, 55
51	ExprRange	lambda_map: 56 start_index: 61 end_index: 59
52	ExprRange	lambda_map: 57 start_index: 61 end_index: 62
53	ExprRange	lambda_map: 58 start_index: 61 end_index: 59
54	Variable
55	ExprRange	lambda_map: 60 start_index: 61 end_index: 62
56	Lambda	parameter: 72 body: 63
57	Lambda	parameter: 72 body: 64
58	Lambda	parameter: 72 body: 65
59	Variable
60	Lambda	parameter: 72 body: 66
61	Literal
62	Variable
63	IndexedVar	variable: 67 index: 72
64	IndexedVar	variable: 68 index: 72
65	IndexedVar	variable: 69 index: 72
66	IndexedVar	variable: 70 index: 72
67	Variable
68	Variable
69	Variable
70	Variable
71	ExprTuple	72
72	Variable