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
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

# build up the expression from sub-expressions
sub_expr1 = [U_1_to_i, V, W_1_to_k]
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()

# 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, 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}

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