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, 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, d_1_to_i, e_1_to_k
from proveit.linear_algebra import TensorProd, VecSpaces
from proveit.logic import And, Equals, Forall, Implies, 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, c_1_to_k, d_1_to_i, e_1_to_k], instance_expr = 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), domains = [U_1_to_i, W_1_to_k, U_1_to_i, 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),\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)}~\\
\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)\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: 18 operand: 5
2	Operation	operator: 20 operands: 4
3	ExprTuple	5
4	ExprTuple	6, 7, 8
5	Lambda	parameters: 9 body: 10
6	ExprRange	lambda_map: 11 start_index: 83 end_index: 81
7	Operation	operator: 23 operands: 12
8	ExprRange	lambda_map: 13 start_index: 83 end_index: 84
9	ExprTuple	73, 74, 75, 77
10	Conditional	value: 14 condition: 15
11	Lambda	parameter: 94 body: 16
12	ExprTuple	56, 30
13	Lambda	parameter: 94 body: 17
14	Operation	operator: 18 operand: 25
15	Operation	operator: 20 operands: 21
16	Operation	operator: 23 operands: 22
17	Operation	operator: 23 operands: 24
18	Literal
19	ExprTuple	25
20	Literal
21	ExprTuple	26, 27, 28, 29
22	ExprTuple	57, 30
23	Literal
24	ExprTuple	58, 30
25	Lambda	parameter: 76 body: 32
26	ExprRange	lambda_map: 33 start_index: 83 end_index: 81
27	ExprRange	lambda_map: 34 start_index: 83 end_index: 84
28	ExprRange	lambda_map: 35 start_index: 83 end_index: 81
29	ExprRange	lambda_map: 36 start_index: 83 end_index: 84
30	Operation	operator: 37 operand: 45
31	ExprTuple	76
32	Conditional	value: 39 condition: 40
33	Lambda	parameter: 94 body: 41
34	Lambda	parameter: 94 body: 42
35	Lambda	parameter: 94 body: 43
36	Lambda	parameter: 94 body: 44
37	Literal
38	ExprTuple	45
39	Operation	operator: 46 operands: 47
40	Operation	operator: 52 operands: 48
41	Operation	operator: 52 operands: 49
42	Operation	operator: 52 operands: 50
43	Operation	operator: 52 operands: 51
44	Operation	operator: 52 operands: 53
45	Variable
46	Literal
47	ExprTuple	54, 55
48	ExprTuple	76, 56
49	ExprTuple	85, 57
50	ExprTuple	86, 58
51	ExprTuple	87, 57
52	Literal
53	ExprTuple	88, 58
54	Operation	operator: 60 operands: 59
55	Operation	operator: 60 operands: 61
56	Variable
57	IndexedVar	variable: 62 index: 94
58	IndexedVar	variable: 63 index: 94
59	ExprTuple	64, 65
60	Literal
61	ExprTuple	66, 67
62	Variable
63	Variable
64	Operation	operator: 71 operands: 68
65	Operation	operator: 71 operands: 69
66	Operation	operator: 71 operands: 70
67	Operation	operator: 71 operands: 72
68	ExprTuple	73, 74
69	ExprTuple	75, 77
70	ExprTuple	73, 76, 74
71	Literal
72	ExprTuple	75, 76, 77
73	ExprRange	lambda_map: 78 start_index: 83 end_index: 81
74	ExprRange	lambda_map: 79 start_index: 83 end_index: 84
75	ExprRange	lambda_map: 80 start_index: 83 end_index: 81
76	Variable
77	ExprRange	lambda_map: 82 start_index: 83 end_index: 84
78	Lambda	parameter: 94 body: 85
79	Lambda	parameter: 94 body: 86
80	Lambda	parameter: 94 body: 87
81	Variable
82	Lambda	parameter: 94 body: 88
83	Literal
84	Variable
85	IndexedVar	variable: 89 index: 94
86	IndexedVar	variable: 90 index: 94
87	IndexedVar	variable: 91 index: 94
88	IndexedVar	variable: 92 index: 94
89	Variable
90	Variable
91	Variable
92	Variable
93	ExprTuple	94
94	Variable