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 A, ExprRange, Function, IndexedVar, K, V, Variable, W, n, v
from proveit.core_expr_types import A_1_to_n, V_1_to_n, W_1_to_n, v_1_to_n
from proveit.linear_algebra import LinMap, TensorProd, VecSpaces
from proveit.logic import Equals, Forall
from proveit.numbers import one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Forall(instance_param_or_params = [V_1_to_n, W_1_to_n], instance_expr = Forall(instance_param_or_params = [A_1_to_n], instance_expr = Forall(instance_param_or_params = [v_1_to_n], instance_expr = Equals(Function(TensorProd(A_1_to_n), [TensorProd(v_1_to_n)]), TensorProd(ExprRange(sub_expr1, Function(IndexedVar(A, sub_expr1), [IndexedVar(v, sub_expr1)]), one, n))), domains = [V_1_to_n]).with_wrapping(), domains = [ExprRange(sub_expr1, LinMap(IndexedVar(V, sub_expr1), IndexedVar(W, sub_expr1)), one, n)]).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_{V_{1}, V_{2}, \ldots, V_{n}, W_{1}, W_{2}, \ldots, W_{n} \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\\
\left[\begin{array}{l}\forall_{\left(A_{1} \in \mathcal{L}\left(V_{1}, W_{1}\right)\right), \left(A_{2} \in \mathcal{L}\left(V_{2}, W_{2}\right)\right), \ldots, \left(A_{n} \in \mathcal{L}\left(V_{n}, W_{n}\right)\right)}~\\
\left[\begin{array}{l}\forall_{\left(v_{1} \in V_{1}\right), \left(v_{2} \in V_{2}\right), \ldots, \left(v_{n} \in V_{n}\right)}~\\
\left(\left(A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{n}\right)\left(v_{1} {\otimes}  v_{2} {\otimes}  \ldots {\otimes}  v_{n}\right) = \left(A_{1}\left(v_{1}\right) {\otimes}  A_{2}\left(v_{2}\right) {\otimes}  \ldots {\otimes}  A_{n}\left(v_{n}\right)\right)\right)\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: 23 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	5, 6
4	Conditional	value: 7 condition: 8
5	ExprRange	lambda_map: 9 start_index: 72 end_index: 73
6	ExprRange	lambda_map: 10 start_index: 72 end_index: 73
7	Operation	operator: 23 operand: 13
8	Operation	operator: 42 operands: 12
9	Lambda	parameter: 81 body: 70
10	Lambda	parameter: 81 body: 59
11	ExprTuple	13
12	ExprTuple	14, 15
13	Lambda	parameters: 55 body: 16
14	ExprRange	lambda_map: 17 start_index: 72 end_index: 73
15	ExprRange	lambda_map: 18 start_index: 72 end_index: 73
16	Conditional	value: 19 condition: 20
17	Lambda	parameter: 81 body: 21
18	Lambda	parameter: 81 body: 22
19	Operation	operator: 23 operand: 29
20	Operation	operator: 42 operands: 25
21	Operation	operator: 27 operands: 26
22	Operation	operator: 27 operands: 28
23	Literal
24	ExprTuple	29
25	ExprTuple	30
26	ExprTuple	70, 31
27	Literal
28	ExprTuple	59, 31
29	Lambda	parameters: 62 body: 32
30	ExprRange	lambda_map: 33 start_index: 72 end_index: 73
31	Operation	operator: 34 operand: 39
32	Conditional	value: 36 condition: 37
33	Lambda	parameter: 81 body: 38
34	Literal
35	ExprTuple	39
36	Operation	operator: 40 operands: 41
37	Operation	operator: 42 operands: 43
38	Operation	operator: 64 operands: 44
39	Variable
40	Literal
41	ExprTuple	45, 46
42	Literal
43	ExprTuple	47
44	ExprTuple	74, 48
45	Operation	operator: 49 operand: 56
46	Operation	operator: 61 operands: 51
47	ExprRange	lambda_map: 52 start_index: 72 end_index: 73
48	Operation	operator: 53 operands: 54
49	Operation	operator: 61 operands: 55
50	ExprTuple	56
51	ExprTuple	57
52	Lambda	parameter: 81 body: 58
53	Literal
54	ExprTuple	70, 59
55	ExprTuple	60
56	Operation	operator: 61 operands: 62
57	ExprRange	lambda_map: 63 start_index: 72 end_index: 73
58	Operation	operator: 64 operands: 65
59	IndexedVar	variable: 66 index: 81
60	ExprRange	lambda_map: 67 start_index: 72 end_index: 73
61	Literal
62	ExprTuple	68
63	Lambda	parameter: 81 body: 69
64	Literal
65	ExprTuple	78, 70
66	Variable
67	Lambda	parameter: 81 body: 74
68	ExprRange	lambda_map: 71 start_index: 72 end_index: 73
69	Operation	operator: 74 operand: 78
70	IndexedVar	variable: 76 index: 81
71	Lambda	parameter: 81 body: 78
72	Literal
73	Variable
74	IndexedVar	variable: 77 index: 81
75	ExprTuple	78
76	Variable
77	Variable
78	IndexedVar	variable: 79 index: 81
79	Variable
80	ExprTuple	81
81	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 ¶