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, i, j, 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
from proveit.numbers import Natural

# 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 = [i, j, k], instance_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, TensorProd(b_1_to_j), c_1_to_k), TensorProd(a_1_to_i, b_1_to_j, c_1_to_k)).with_wrapping_at(1), domains = sub_expr1).with_wrapping(), domain = VecSpaces(K)).with_wrapping(), domain = Natural)

# 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())

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