Theorem tensor_prod_factorization_from_summation of type Forall ¶

from the theory of proveit.linear_algebra.tensors ¶

import proveit
# Automation is not needed when only building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_theorem_expr # Load the stored theorem expression as 'stored_expr'
# import the special expression
from proveit.linear_algebra.tensors import tensor_prod_factorization_from_summation

# check that the built expression is the same as the stored expression
assert tensor_prod_factorization_from_summation.expr == stored_expr
assert tensor_prod_factorization_from_summation.expr._style_id == stored_expr._style_id
print("Passed sanity check: tensor_prod_factorization_from_summation matches stored_expr")

Passed sanity check: tensor_prod_factorization_from_summation matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\forall_{K, f, Q}~\left[\forall_{i \in \mathbb{N}, j \in \mathbb{N}^+, k \in \mathbb{N}}~\left[\forall_{V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{i}, c_{1}, c_{2}, \ldots, c_{k}}~\left(\begin{array}{c} \begin{array}{l} \left[\forall_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(\left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} f\left(b_{1}, b_{2}, \ldots, b_{j}\right){\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \in V\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} f\left(b_{1}, b_{2}, \ldots, b_{j}\right){\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right)\right] \\  = \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right]{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \end{array} \end{array}\right) \end{array} \end{array}\right)\right]\right]\right]

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

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

Theorem tensor_prod_factorization_from_summation of type Forall¶

from the theory of proveit.linear_algebra.tensors¶

Theorem tensor_prod_factorization_from_summation of type Forall ¶

from the theory of proveit.linear_algebra.tensors ¶