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, 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
from proveit.numbers import NaturalPos, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Lambda(K, Forall(instance_param_or_params = [n], instance_expr = 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(), domain = 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())

K \mapsto \left[\forall_{n \in \mathbb{N}^+}~\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}\right]\right]

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 48 body: 1
1	Operation	operator: 32 operand: 3
2	ExprTuple	3
3	Lambda	parameter: 82 body: 5
4	ExprTuple	82
5	Conditional	value: 6 condition: 7
6	Operation	operator: 32 operand: 10
7	Operation	operator: 73 operands: 9
8	ExprTuple	10
9	ExprTuple	82, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	14, 15
13	Conditional	value: 16 condition: 17
14	ExprRange	lambda_map: 18 start_index: 81 end_index: 82
15	ExprRange	lambda_map: 19 start_index: 81 end_index: 82
16	Operation	operator: 32 operand: 22
17	Operation	operator: 51 operands: 21
18	Lambda	parameter: 90 body: 79
19	Lambda	parameter: 90 body: 68
20	ExprTuple	22
21	ExprTuple	23, 24
22	Lambda	parameters: 64 body: 25
23	ExprRange	lambda_map: 26 start_index: 81 end_index: 82
24	ExprRange	lambda_map: 27 start_index: 81 end_index: 82
25	Conditional	value: 28 condition: 29
26	Lambda	parameter: 90 body: 30
27	Lambda	parameter: 90 body: 31
28	Operation	operator: 32 operand: 38
29	Operation	operator: 51 operands: 34
30	Operation	operator: 36 operands: 35
31	Operation	operator: 36 operands: 37
32	Literal
33	ExprTuple	38
34	ExprTuple	39
35	ExprTuple	79, 40
36	Literal
37	ExprTuple	68, 40
38	Lambda	parameters: 71 body: 41
39	ExprRange	lambda_map: 42 start_index: 81 end_index: 82
40	Operation	operator: 43 operand: 48
41	Conditional	value: 45 condition: 46
42	Lambda	parameter: 90 body: 47
43	Literal
44	ExprTuple	48
45	Operation	operator: 49 operands: 50
46	Operation	operator: 51 operands: 52
47	Operation	operator: 73 operands: 53
48	Variable
49	Literal
50	ExprTuple	54, 55
51	Literal
52	ExprTuple	56
53	ExprTuple	83, 57
54	Operation	operator: 58 operand: 65
55	Operation	operator: 70 operands: 60
56	ExprRange	lambda_map: 61 start_index: 81 end_index: 82
57	Operation	operator: 62 operands: 63
58	Operation	operator: 70 operands: 64
59	ExprTuple	65
60	ExprTuple	66
61	Lambda	parameter: 90 body: 67
62	Literal
63	ExprTuple	79, 68
64	ExprTuple	69
65	Operation	operator: 70 operands: 71
66	ExprRange	lambda_map: 72 start_index: 81 end_index: 82
67	Operation	operator: 73 operands: 74
68	IndexedVar	variable: 75 index: 90
69	ExprRange	lambda_map: 76 start_index: 81 end_index: 82
70	Literal
71	ExprTuple	77
72	Lambda	parameter: 90 body: 78
73	Literal
74	ExprTuple	87, 79
75	Variable
76	Lambda	parameter: 90 body: 83
77	ExprRange	lambda_map: 80 start_index: 81 end_index: 82
78	Operation	operator: 83 operand: 87
79	IndexedVar	variable: 85 index: 90
80	Lambda	parameter: 90 body: 87
81	Literal
82	Variable
83	IndexedVar	variable: 86 index: 90
84	ExprTuple	87
85	Variable
86	Variable
87	IndexedVar	variable: 88 index: 90
88	Variable
89	ExprTuple	90
90	Variable