Expression of type Forall ¶

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 K
from proveit.core_expr_types import U_1_to_i, V_1_to_j, W_1_to_k, a_1_to_i, b_1_to_j, c_1_to_k
from proveit.linear_algebra import TensorProd, VecSpaces
from proveit.logic import Equals, Forall

# build up the expression from sub-expressions
sub_expr1 = [U_1_to_i, V_1_to_j, W_1_to_k]
expr = Forall(instance_param_or_params = sub_expr1, instance_expr = Forall(instance_param_or_params = [a_1_to_i, b_1_to_j, c_1_to_k], instance_expr = Equals(TensorProd(a_1_to_i, b_1_to_j, c_1_to_k), TensorProd(a_1_to_i, TensorProd(b_1_to_j), c_1_to_k)).with_wrapping_at(2), domains = sub_expr1).with_wrapping(), domain = VecSpaces(K)).with_wrapping()

# 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())

\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}

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

Expression of type Forall¶

from the theory of proveit.linear_algebra.tensors¶

Expression of type Forall ¶

from the theory of proveit.linear_algebra.tensors ¶