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, f, i, k
from proveit.core_expr_types import a_1_to_i, c_1_to_k, d_1_to_i, f_1_to_k
from proveit.linear_algebra import TensorProd, VecZero
from proveit.logic import And, Equals, Forall, Implies, InSet, NotEquals
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, e], instance_expr = Implies(Equals(TensorProd(a_1_to_i, b, c_1_to_k), TensorProd(d_1_to_i, e, f_1_to_k)).with_wrapping_at(2), Equals(TensorProd(a_1_to_i, c_1_to_k), TensorProd(d_1_to_i, f_1_to_k)).with_wrapping_at(2)).with_wrapping_at(2), domain = V, conditions = [Equals(b, e), NotEquals(b, VecZero(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(f, 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, e \in V~|~b = e, b \neq \vec{0}\left(V\right)}~\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} 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} e{\otimes} f_{1} {\otimes}  f_{2} {\otimes}  \ldots {\otimes}  f_{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} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) =  \\ \left(d_{1} {\otimes}  d_{2} {\otimes}  \ldots {\otimes}  d_{i}{\otimes} f_{1} {\otimes}  f_{2} {\otimes}  \ldots {\otimes}  f_{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(f_{1} \in W_{1}\right) ,  \left(f_{2} \in W_{2}\right) ,  \ldots ,  \left(f_{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: 6
2	Operation	operator: 24 operands: 5
3	Literal
4	ExprTuple	6
5	ExprTuple	7, 8, 9, 10
6	Lambda	parameters: 44 body: 11
7	ExprRange	lambda_map: 12 start_index: 73 end_index: 71
8	ExprRange	lambda_map: 13 start_index: 73 end_index: 74
9	ExprRange	lambda_map: 14 start_index: 73 end_index: 71
10	ExprRange	lambda_map: 15 start_index: 73 end_index: 74
11	Conditional	value: 16 condition: 17
12	Lambda	parameter: 84 body: 18
13	Lambda	parameter: 84 body: 19
14	Lambda	parameter: 84 body: 20
15	Lambda	parameter: 84 body: 21
16	Operation	operator: 22 operands: 23
17	Operation	operator: 24 operands: 25
18	Operation	operator: 41 operands: 26
19	Operation	operator: 41 operands: 27
20	Operation	operator: 41 operands: 28
21	Operation	operator: 41 operands: 29
22	Literal
23	ExprTuple	30, 31
24	Literal
25	ExprTuple	32, 33, 34, 35
26	ExprTuple	75, 36
27	ExprTuple	76, 37
28	ExprTuple	77, 36
29	ExprTuple	78, 37
30	Operation	operator: 43 operands: 38
31	Operation	operator: 43 operands: 39
32	Operation	operator: 41 operands: 40
33	Operation	operator: 41 operands: 42
34	Operation	operator: 43 operands: 44
35	Operation	operator: 45 operands: 46
36	IndexedVar	variable: 47 index: 84
37	IndexedVar	variable: 48 index: 84
38	ExprTuple	49, 50
39	ExprTuple	51, 52
40	ExprTuple	61, 67
41	Literal
42	ExprTuple	62, 67
43	Literal
44	ExprTuple	61, 62
45	Literal
46	ExprTuple	61, 53
47	Variable
48	Variable
49	Operation	operator: 57 operands: 54
50	Operation	operator: 57 operands: 55
51	Operation	operator: 57 operands: 56
52	Operation	operator: 57 operands: 58
53	Operation	operator: 59 operand: 67
54	ExprTuple	63, 61, 64
55	ExprTuple	65, 62, 66
56	ExprTuple	63, 64
57	Literal
58	ExprTuple	65, 66
59	Literal
60	ExprTuple	67
61	Variable
62	Variable
63	ExprRange	lambda_map: 68 start_index: 73 end_index: 71
64	ExprRange	lambda_map: 69 start_index: 73 end_index: 74
65	ExprRange	lambda_map: 70 start_index: 73 end_index: 71
66	ExprRange	lambda_map: 72 start_index: 73 end_index: 74
67	Variable
68	Lambda	parameter: 84 body: 75
69	Lambda	parameter: 84 body: 76
70	Lambda	parameter: 84 body: 77
71	Variable
72	Lambda	parameter: 84 body: 78
73	Literal
74	Variable
75	IndexedVar	variable: 79 index: 84
76	IndexedVar	variable: 80 index: 84
77	IndexedVar	variable: 81 index: 84
78	IndexedVar	variable: 82 index: 84
79	Variable
80	Variable
81	Variable
82	Variable
83	ExprTuple	84
84	Variable