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, V, alpha, b, i, k
from proveit.core_expr_types import U_1_to_i, W_1_to_k, a_1_to_i, c_1_to_k
from proveit.linear_algebra import ScalarMult, 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, W_1_to_k]
expr = Forall(instance_param_or_params = [i, k], instance_expr = Forall(instance_param_or_params = sub_expr1, instance_expr = Forall(instance_param_or_params = [a_1_to_i, b, c_1_to_k], instance_expr = Equals(TensorProd(a_1_to_i, ScalarMult(alpha, b), c_1_to_k), ScalarMult(alpha, TensorProd(a_1_to_i, b, 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, k \in \mathbb{N}}~\left[\begin{array}{l}\forall_{U_{1}, U_{2}, \ldots, U_{i}, V, 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), b \in V,\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(\alpha \cdot b\right){\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \\  = \left(\alpha \cdot \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} b{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right)\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: 23 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	78, 81
4	Conditional	value: 5 condition: 6
5	Operation	operator: 23 operand: 9
6	Operation	operator: 40 operands: 8
7	ExprTuple	9
8	ExprTuple	10, 11
9	Lambda	parameters: 12 body: 13
10	Operation	operator: 69 operands: 14
11	Operation	operator: 69 operands: 15
12	ExprTuple	16, 61, 17
13	Conditional	value: 18 condition: 19
14	ExprTuple	78, 20
15	ExprTuple	81, 20
16	ExprRange	lambda_map: 21 start_index: 80 end_index: 78
17	ExprRange	lambda_map: 22 start_index: 80 end_index: 81
18	Operation	operator: 23 operand: 26
19	Operation	operator: 40 operands: 25
20	Literal
21	Lambda	parameter: 89 body: 75
22	Lambda	parameter: 89 body: 76
23	Literal
24	ExprTuple	26
25	ExprTuple	27, 28, 29
26	Lambda	parameters: 67 body: 30
27	ExprRange	lambda_map: 31 start_index: 80 end_index: 78
28	Operation	operator: 43 operands: 32
29	ExprRange	lambda_map: 33 start_index: 80 end_index: 81
30	Conditional	value: 34 condition: 35
31	Lambda	parameter: 89 body: 36
32	ExprTuple	61, 50
33	Lambda	parameter: 89 body: 37
34	Operation	operator: 38 operands: 39
35	Operation	operator: 40 operands: 41
36	Operation	operator: 43 operands: 42
37	Operation	operator: 43 operands: 44
38	Literal
39	ExprTuple	45, 46
40	Literal
41	ExprTuple	47, 48, 49
42	ExprTuple	75, 50
43	Literal
44	ExprTuple	76, 50
45	Operation	operator: 66 operands: 51
46	Operation	operator: 64 operands: 52
47	ExprRange	lambda_map: 53 start_index: 80 end_index: 78
48	Operation	operator: 69 operands: 54
49	ExprRange	lambda_map: 55 start_index: 80 end_index: 81
50	Operation	operator: 56 operand: 63
51	ExprTuple	72, 58, 74
52	ExprTuple	71, 59
53	Lambda	parameter: 89 body: 60
54	ExprTuple	73, 61
55	Lambda	parameter: 89 body: 62
56	Literal
57	ExprTuple	63
58	Operation	operator: 64 operands: 65
59	Operation	operator: 66 operands: 67
60	Operation	operator: 69 operands: 68
61	Variable
62	Operation	operator: 69 operands: 70
63	Variable
64	Literal
65	ExprTuple	71, 73
66	Literal
67	ExprTuple	72, 73, 74
68	ExprTuple	84, 75
69	Literal
70	ExprTuple	85, 76
71	Variable
72	ExprRange	lambda_map: 77 start_index: 80 end_index: 78
73	Variable
74	ExprRange	lambda_map: 79 start_index: 80 end_index: 81
75	IndexedVar	variable: 82 index: 89
76	IndexedVar	variable: 83 index: 89
77	Lambda	parameter: 89 body: 84
78	Variable
79	Lambda	parameter: 89 body: 85
80	Literal
81	Variable
82	Variable
83	Variable
84	IndexedVar	variable: 86 index: 89
85	IndexedVar	variable: 87 index: 89
86	Variable
87	Variable
88	ExprTuple	89
89	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 ¶