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 Equals, Forall, InSet
from proveit.numbers import NaturalPos, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Lambda(n, Conditional(Forall(instance_param_or_params = [V_1_to_n, W_1_to_n], instance_expr = 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(IndexedVar(V, sub_expr1), IndexedVar(W, sub_expr1)), one, n)]).with_wrapping(), domain = VecSpaces(K)).with_wrapping(), InSet(n, NaturalPos)))

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())

n \mapsto \left\{\begin{array}{l}\forall_{V_{1}, V_{2}, \ldots, V_{n}, W_{1}, W_{2}, \ldots, W_{n} \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\\
\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}\right]\end{array} \textrm{ if } n \in \mathbb{N}^+\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 79 body: 2
1	ExprTuple	79
2	Conditional	value: 3 condition: 4
3	Operation	operator: 29 operand: 7
4	Operation	operator: 70 operands: 6
5	ExprTuple	7
6	ExprTuple	79, 8
7	Lambda	parameters: 9 body: 10
8	Literal
9	ExprTuple	11, 12
10	Conditional	value: 13 condition: 14
11	ExprRange	lambda_map: 15 start_index: 78 end_index: 79
12	ExprRange	lambda_map: 16 start_index: 78 end_index: 79
13	Operation	operator: 29 operand: 19
14	Operation	operator: 48 operands: 18
15	Lambda	parameter: 87 body: 76
16	Lambda	parameter: 87 body: 65
17	ExprTuple	19
18	ExprTuple	20, 21
19	Lambda	parameters: 61 body: 22
20	ExprRange	lambda_map: 23 start_index: 78 end_index: 79
21	ExprRange	lambda_map: 24 start_index: 78 end_index: 79
22	Conditional	value: 25 condition: 26
23	Lambda	parameter: 87 body: 27
24	Lambda	parameter: 87 body: 28
25	Operation	operator: 29 operand: 35
26	Operation	operator: 48 operands: 31
27	Operation	operator: 33 operands: 32
28	Operation	operator: 33 operands: 34
29	Literal
30	ExprTuple	35
31	ExprTuple	36
32	ExprTuple	76, 37
33	Literal
34	ExprTuple	65, 37
35	Lambda	parameters: 68 body: 38
36	ExprRange	lambda_map: 39 start_index: 78 end_index: 79
37	Operation	operator: 40 operand: 45
38	Conditional	value: 42 condition: 43
39	Lambda	parameter: 87 body: 44
40	Literal
41	ExprTuple	45
42	Operation	operator: 46 operands: 47
43	Operation	operator: 48 operands: 49
44	Operation	operator: 70 operands: 50
45	Variable
46	Literal
47	ExprTuple	51, 52
48	Literal
49	ExprTuple	53
50	ExprTuple	80, 54
51	Operation	operator: 55 operand: 62
52	Operation	operator: 67 operands: 57
53	ExprRange	lambda_map: 58 start_index: 78 end_index: 79
54	Operation	operator: 59 operands: 60
55	Operation	operator: 67 operands: 61
56	ExprTuple	62
57	ExprTuple	63
58	Lambda	parameter: 87 body: 64
59	Literal
60	ExprTuple	76, 65
61	ExprTuple	66
62	Operation	operator: 67 operands: 68
63	ExprRange	lambda_map: 69 start_index: 78 end_index: 79
64	Operation	operator: 70 operands: 71
65	IndexedVar	variable: 72 index: 87
66	ExprRange	lambda_map: 73 start_index: 78 end_index: 79
67	Literal
68	ExprTuple	74
69	Lambda	parameter: 87 body: 75
70	Literal
71	ExprTuple	84, 76
72	Variable
73	Lambda	parameter: 87 body: 80
74	ExprRange	lambda_map: 77 start_index: 78 end_index: 79
75	Operation	operator: 80 operand: 84
76	IndexedVar	variable: 82 index: 87
77	Lambda	parameter: 87 body: 84
78	Literal
79	Variable
80	IndexedVar	variable: 83 index: 87
81	ExprTuple	84
82	Variable
83	Variable
84	IndexedVar	variable: 85 index: 87
85	Variable
86	ExprTuple	87
87	Variable