Expression of type Lambda

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 A, Conditional, ExprRange, Function, IndexedVar, K, Lambda, V, Variable, W, n, v
from proveit.core_expr_types import A_1_to_n, V_1_to_n, W_1_to_n, v_1_to_n
from proveit.linear_algebra import LinMap, 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 = IndexedVar(V, sub_expr1)
sub_expr3 = IndexedVar(W, sub_expr1)
sub_expr4 = VecSpaces(K)
expr = Lambda([V_1_to_n, W_1_to_n], Conditional(Forall(instance_param_or_params = [A_1_to_n], instance_expr = Forall(instance_param_or_params = [v_1_to_n], instance_expr = Equals(Function(TensorProd(A_1_to_n), [TensorProd(v_1_to_n)]), TensorProd(ExprRange(sub_expr1, Function(IndexedVar(A, sub_expr1), [IndexedVar(v, sub_expr1)]), one, n))), domains = [V_1_to_n]).with_wrapping(), domains = [ExprRange(sub_expr1, LinMap(sub_expr2, sub_expr3), one, n)]).with_wrapping(), And(ExprRange(sub_expr1, InClass(sub_expr2, sub_expr4), one, n), ExprRange(sub_expr1, InClass(sub_expr3, sub_expr4), one, n))))

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(V_{1}, V_{2}, \ldots, V_{n}, W_{1}, W_{2}, \ldots, W_{n}\right) \mapsto \left\{\begin{array}{l}\forall_{\left(A_{1} \in \mathcal{L}\left(V_{1}, W_{1}\right)\right), \left(A_{2} \in \mathcal{L}\left(V_{2}, W_{2}\right)\right), \ldots, \left(A_{n} \in \mathcal{L}\left(V_{n}, W_{n}\right)\right)}~\\
\left[\begin{array}{l}\forall_{\left(v_{1} \in V_{1}\right), \left(v_{2} \in V_{2}\right), \ldots, \left(v_{n} \in V_{n}\right)}~\\
\left(\left(A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{n}\right)\left(v_{1} {\otimes}  v_{2} {\otimes}  \ldots {\otimes}  v_{n}\right) = \left(A_{1}\left(v_{1}\right) {\otimes}  A_{2}\left(v_{2}\right) {\otimes}  \ldots {\otimes}  A_{n}\left(v_{n}\right)\right)\right)\end{array}\right]\end{array} \textrm{ if } \left(V_{1} \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)\right) ,  \left(V_{2} \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)\right) ,  \ldots ,  \left(V_{n} \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)\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_{n} \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)\right)\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameters: 1 body: 2
1	ExprTuple	3, 4
2	Conditional	value: 5 condition: 6
3	ExprRange	lambda_map: 7 start_index: 70 end_index: 71
4	ExprRange	lambda_map: 8 start_index: 70 end_index: 71
5	Operation	operator: 21 operand: 11
6	Operation	operator: 40 operands: 10
7	Lambda	parameter: 79 body: 68
8	Lambda	parameter: 79 body: 57
9	ExprTuple	11
10	ExprTuple	12, 13
11	Lambda	parameters: 53 body: 14
12	ExprRange	lambda_map: 15 start_index: 70 end_index: 71
13	ExprRange	lambda_map: 16 start_index: 70 end_index: 71
14	Conditional	value: 17 condition: 18
15	Lambda	parameter: 79 body: 19
16	Lambda	parameter: 79 body: 20
17	Operation	operator: 21 operand: 27
18	Operation	operator: 40 operands: 23
19	Operation	operator: 25 operands: 24
20	Operation	operator: 25 operands: 26
21	Literal
22	ExprTuple	27
23	ExprTuple	28
24	ExprTuple	68, 29
25	Literal
26	ExprTuple	57, 29
27	Lambda	parameters: 60 body: 30
28	ExprRange	lambda_map: 31 start_index: 70 end_index: 71
29	Operation	operator: 32 operand: 37
30	Conditional	value: 34 condition: 35
31	Lambda	parameter: 79 body: 36
32	Literal
33	ExprTuple	37
34	Operation	operator: 38 operands: 39
35	Operation	operator: 40 operands: 41
36	Operation	operator: 62 operands: 42
37	Variable
38	Literal
39	ExprTuple	43, 44
40	Literal
41	ExprTuple	45
42	ExprTuple	72, 46
43	Operation	operator: 47 operand: 54
44	Operation	operator: 59 operands: 49
45	ExprRange	lambda_map: 50 start_index: 70 end_index: 71
46	Operation	operator: 51 operands: 52
47	Operation	operator: 59 operands: 53
48	ExprTuple	54
49	ExprTuple	55
50	Lambda	parameter: 79 body: 56
51	Literal
52	ExprTuple	68, 57
53	ExprTuple	58
54	Operation	operator: 59 operands: 60
55	ExprRange	lambda_map: 61 start_index: 70 end_index: 71
56	Operation	operator: 62 operands: 63
57	IndexedVar	variable: 64 index: 79
58	ExprRange	lambda_map: 65 start_index: 70 end_index: 71
59	Literal
60	ExprTuple	66
61	Lambda	parameter: 79 body: 67
62	Literal
63	ExprTuple	76, 68
64	Variable
65	Lambda	parameter: 79 body: 72
66	ExprRange	lambda_map: 69 start_index: 70 end_index: 71
67	Operation	operator: 72 operand: 76
68	IndexedVar	variable: 74 index: 79
69	Lambda	parameter: 79 body: 76
70	Literal
71	Variable
72	IndexedVar	variable: 75 index: 79
73	ExprTuple	76
74	Variable
75	Variable
76	IndexedVar	variable: 77 index: 79
77	Variable
78	ExprTuple	79
79	Variable