Theorem tensor_prod_factorization_from_summation_with_scalar_mult 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_with_scalar_mult

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

Passed sanity check: tensor_prod_factorization_from_summation_with_scalar_mult matches stored_expr

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

\forall_{K, f, Q, s}~\left[\forall_{i \in \mathbb{N}, j \in \mathbb{N}^+, k \in \mathbb{N}}~\left[\begin{array}{l}\forall_{V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\\
\left[\begin{array}{l}\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} \left(s\left(b_{1}, b_{2}, \ldots, b_{j}\right) \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\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(s\left(b_{1}, b_{2}, \ldots, b_{j}\right) \cdot \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)\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)}~\left(s\left(b_{1}, b_{2}, \ldots, b_{j}\right) \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right)\right]{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \end{array} \end{array}\right) \end{array} \end{array}\right)\end{array}\right]\end{array}\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, 83, 73, 82
4	Operation	operator: 41 operand: 6
5	ExprTuple	6
6	Lambda	parameters: 7 body: 8
7	ExprTuple	79, 92, 81
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: 60 body: 19
15	Operation	operator: 54 operands: 20
16	Operation	operator: 54 operands: 21
17	Operation	operator: 54 operands: 22
18	ExprTuple	60
19	Conditional	value: 23 condition: 24
20	ExprTuple	79, 26
21	ExprTuple	92, 25
22	ExprTuple	81, 26
23	Operation	operator: 41 operand: 30
24	Operation	operator: 28 operands: 29
25	Literal
26	Literal
27	ExprTuple	30
28	Literal
29	ExprTuple	60, 31
30	Lambda	parameters: 32 body: 33
31	Operation	operator: 34 operand: 38
32	ExprTuple	74, 75
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: 84 body: 48
46	Operation	operator: 57 operand: 52
47	Operation	operator: 69 operands: 50
48	Conditional	value: 51 condition: 68
49	ExprTuple	52
50	ExprTuple	74, 53, 75
51	Operation	operator: 54 operands: 55
52	Lambda	parameters: 84 body: 56
53	Operation	operator: 57 operand: 62
54	Literal
55	ExprTuple	59, 60
56	Conditional	value: 61 condition: 68
57	Literal
58	ExprTuple	62
59	Operation	operator: 69 operands: 63
60	Variable
61	Operation	operator: 71 operands: 64
62	Lambda	parameters: 84 body: 65
63	ExprTuple	74, 67, 75
64	ExprTuple	76, 66
65	Conditional	value: 67 condition: 68
66	Operation	operator: 69 operands: 70
67	Operation	operator: 71 operands: 72
68	Operation	operator: 73 operands: 84
69	Literal
70	ExprTuple	74, 77, 75
71	Literal
72	ExprTuple	76, 77
73	Variable
74	ExprRange	lambda_map: 78 start_index: 91 end_index: 79
75	ExprRange	lambda_map: 80 start_index: 91 end_index: 81
76	Operation	operator: 82 operands: 84
77	Operation	operator: 83 operands: 84
78	Lambda	parameter: 96 body: 85
79	Variable
80	Lambda	parameter: 96 body: 86
81	Variable
82	Variable
83	Variable
84	ExprTuple	87
85	IndexedVar	variable: 88 index: 96
86	IndexedVar	variable: 89 index: 96
87	ExprRange	lambda_map: 90 start_index: 91 end_index: 92
88	Variable
89	Variable
90	Lambda	parameter: 96 body: 93
91	Literal
92	Variable
93	IndexedVar	variable: 94 index: 96
94	Variable
95	ExprTuple	96
96	Variable

Theorem tensor_prod_factorization_from_summation_with_scalar_mult of type Forall¶

from the theory of proveit.linear_algebra.tensors¶

Theorem tensor_prod_factorization_from_summation_with_scalar_mult of type Forall ¶

from the theory of proveit.linear_algebra.tensors ¶