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 NaturalPos, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Forall(instance_param_or_params = [n], instance_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(), domain = NaturalPos)

# 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_{n \in \mathbb{N}^+}~\left[\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}\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: 31 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 81 body: 4
3	ExprTuple	81
4	Conditional	value: 5 condition: 6
5	Operation	operator: 31 operand: 9
6	Operation	operator: 72 operands: 8
7	ExprTuple	9
8	ExprTuple	81, 10
9	Lambda	parameters: 11 body: 12
10	Literal
11	ExprTuple	13, 14
12	Conditional	value: 15 condition: 16
13	ExprRange	lambda_map: 17 start_index: 80 end_index: 81
14	ExprRange	lambda_map: 18 start_index: 80 end_index: 81
15	Operation	operator: 31 operand: 21
16	Operation	operator: 50 operands: 20
17	Lambda	parameter: 89 body: 78
18	Lambda	parameter: 89 body: 67
19	ExprTuple	21
20	ExprTuple	22, 23
21	Lambda	parameters: 63 body: 24
22	ExprRange	lambda_map: 25 start_index: 80 end_index: 81
23	ExprRange	lambda_map: 26 start_index: 80 end_index: 81
24	Conditional	value: 27 condition: 28
25	Lambda	parameter: 89 body: 29
26	Lambda	parameter: 89 body: 30
27	Operation	operator: 31 operand: 37
28	Operation	operator: 50 operands: 33
29	Operation	operator: 35 operands: 34
30	Operation	operator: 35 operands: 36
31	Literal
32	ExprTuple	37
33	ExprTuple	38
34	ExprTuple	78, 39
35	Literal
36	ExprTuple	67, 39
37	Lambda	parameters: 70 body: 40
38	ExprRange	lambda_map: 41 start_index: 80 end_index: 81
39	Operation	operator: 42 operand: 47
40	Conditional	value: 44 condition: 45
41	Lambda	parameter: 89 body: 46
42	Literal
43	ExprTuple	47
44	Operation	operator: 48 operands: 49
45	Operation	operator: 50 operands: 51
46	Operation	operator: 72 operands: 52
47	Variable
48	Literal
49	ExprTuple	53, 54
50	Literal
51	ExprTuple	55
52	ExprTuple	82, 56
53	Operation	operator: 57 operand: 64
54	Operation	operator: 69 operands: 59
55	ExprRange	lambda_map: 60 start_index: 80 end_index: 81
56	Operation	operator: 61 operands: 62
57	Operation	operator: 69 operands: 63
58	ExprTuple	64
59	ExprTuple	65
60	Lambda	parameter: 89 body: 66
61	Literal
62	ExprTuple	78, 67
63	ExprTuple	68
64	Operation	operator: 69 operands: 70
65	ExprRange	lambda_map: 71 start_index: 80 end_index: 81
66	Operation	operator: 72 operands: 73
67	IndexedVar	variable: 74 index: 89
68	ExprRange	lambda_map: 75 start_index: 80 end_index: 81
69	Literal
70	ExprTuple	76
71	Lambda	parameter: 89 body: 77
72	Literal
73	ExprTuple	86, 78
74	Variable
75	Lambda	parameter: 89 body: 82
76	ExprRange	lambda_map: 79 start_index: 80 end_index: 81
77	Operation	operator: 82 operand: 86
78	IndexedVar	variable: 84 index: 89
79	Lambda	parameter: 89 body: 86
80	Literal
81	Variable
82	IndexedVar	variable: 85 index: 89
83	ExprTuple	86
84	Variable
85	Variable
86	IndexedVar	variable: 87 index: 89
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 ¶