Theorem tensor_prod_association of type Forall ¶

from the theory of proveit.linear_algebra.tensors ¶

In [1]:

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_association

In [2]:

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

Passed sanity check: tensor_prod_association matches stored_expr

In [3]:

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

\forall_{K}~\left[\forall_{i, j, k \in \mathbb{N}}~\left[\begin{array}{l}\forall_{U_{1}, U_{2}, \ldots, U_{i}, V_{1}, V_{2}, \ldots, V_{j}, W_{1}, W_{2}, \ldots, W_{k} \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\\
\left[\begin{array}{l}\forall_{\left(a_{1} \in U_{1}\right), \left(a_{2} \in U_{2}\right), \ldots, \left(a_{i} \in U_{i}\right),\left(b_{1} \in V_{1}\right), \left(b_{2} \in V_{2}\right), \ldots, \left(b_{j} \in V_{j}\right),\left(c_{1} \in W_{1}\right), \left(c_{2} \in W_{2}\right), \ldots, \left(c_{k} \in W_{k}\right)}~\\
\left(\begin{array}{c} \begin{array}{l} \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i}{\otimes} b_{1} {\otimes}  b_{2} {\otimes}  \ldots {\otimes}  b_{j}{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) =  \\ \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} \left(b_{1} {\otimes}  b_{2} {\otimes}  \ldots {\otimes}  b_{j}\right){\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \end{array} \end{array}\right)\end{array}\right]\end{array}\right]\right]

In [4]:

stored_expr.style_options()

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',)

In [5]:

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions
0	Operation	operator: 30 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 73 body: 3
3	Operation	operator: 30 operand: 5
4	ExprTuple	5
5	Lambda	parameters: 6 body: 7
6	ExprTuple	75, 92, 79
7	Conditional	value: 8 condition: 9
8	Operation	operator: 30 operand: 12
9	Operation	operator: 48 operands: 11
10	ExprTuple	12
11	ExprTuple	13, 14, 15
12	Lambda	parameters: 16 body: 17
13	Operation	operator: 82 operands: 18
14	Operation	operator: 82 operands: 19
15	Operation	operator: 82 operands: 20
16	ExprTuple	21, 22, 23
17	Conditional	value: 24 condition: 25
18	ExprTuple	75, 26
19	ExprTuple	92, 26
20	ExprTuple	79, 26
21	ExprRange	lambda_map: 27 start_index: 91 end_index: 75
22	ExprRange	lambda_map: 28 start_index: 91 end_index: 92
23	ExprRange	lambda_map: 29 start_index: 91 end_index: 79
24	Operation	operator: 30 operand: 33
25	Operation	operator: 48 operands: 32
26	Literal
27	Lambda	parameter: 101 body: 86
28	Lambda	parameter: 101 body: 87
29	Lambda	parameter: 101 body: 89
30	Literal
31	ExprTuple	33
32	ExprTuple	34, 35, 36
33	Lambda	parameters: 60 body: 37
34	ExprRange	lambda_map: 38 start_index: 91 end_index: 75
35	ExprRange	lambda_map: 39 start_index: 91 end_index: 92
36	ExprRange	lambda_map: 40 start_index: 91 end_index: 79
37	Conditional	value: 41 condition: 42
38	Lambda	parameter: 101 body: 43
39	Lambda	parameter: 101 body: 44
40	Lambda	parameter: 101 body: 45
41	Operation	operator: 46 operands: 47
42	Operation	operator: 48 operands: 49
43	Operation	operator: 52 operands: 50
44	Operation	operator: 52 operands: 51
45	Operation	operator: 52 operands: 53
46	Literal
47	ExprTuple	54, 55
48	Literal
49	ExprTuple	56, 57, 58
50	ExprTuple	86, 59
51	ExprTuple	87, 59
52	Literal
53	ExprTuple	89, 59
54	Operation	operator: 76 operands: 60
55	Operation	operator: 76 operands: 61
56	ExprRange	lambda_map: 62 start_index: 91 end_index: 75
57	ExprRange	lambda_map: 63 start_index: 91 end_index: 92
58	ExprRange	lambda_map: 64 start_index: 91 end_index: 79
59	Operation	operator: 65 operand: 73
60	ExprTuple	67, 84, 69
61	ExprTuple	67, 68, 69
62	Lambda	parameter: 101 body: 70
63	Lambda	parameter: 101 body: 71
64	Lambda	parameter: 101 body: 72
65	Literal
66	ExprTuple	73
67	ExprRange	lambda_map: 74 start_index: 91 end_index: 75
68	Operation	operator: 76 operands: 77
69	ExprRange	lambda_map: 78 start_index: 91 end_index: 79
70	Operation	operator: 82 operands: 80
71	Operation	operator: 82 operands: 81
72	Operation	operator: 82 operands: 83
73	Variable
74	Lambda	parameter: 101 body: 85
75	Variable
76	Literal
77	ExprTuple	84
78	Lambda	parameter: 101 body: 88
79	Variable
80	ExprTuple	85, 86
81	ExprTuple	98, 87
82	Literal
83	ExprTuple	88, 89
84	ExprRange	lambda_map: 90 start_index: 91 end_index: 92
85	IndexedVar	variable: 93 index: 101
86	IndexedVar	variable: 94 index: 101
87	IndexedVar	variable: 95 index: 101
88	IndexedVar	variable: 96 index: 101
89	IndexedVar	variable: 97 index: 101
90	Lambda	parameter: 101 body: 98
91	Literal
92	Variable
93	Variable
94	Variable
95	Variable
96	Variable
97	Variable
98	IndexedVar	variable: 99 index: 101
99	Variable
100	ExprTuple	101
101	Variable