Expression of type Forall ¶

from the theory of proveit.linear_algebra.inner_products ¶

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, B, ExprRange, IndexedVar, Variable, i, lambda_, m, n, v
from proveit.core_expr_types import a_1_to_m, a_i, b_1_to_n, b_i, lambda_i
from proveit.linear_algebra import OrthoNormBases, ScalarMult, TensorProd, VecSum
from proveit.logic import Equals, Exists, Forall, InSet, Set
from proveit.numbers import Interval, Min, RealNonNeg, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Min(m, n)
expr = Forall(instance_param_or_params = [v], instance_expr = Exists(instance_param_or_params = [a_1_to_m], instance_expr = Exists(instance_param_or_params = [b_1_to_n], instance_expr = Exists(instance_param_or_params = [ExprRange(sub_expr1, IndexedVar(lambda_, sub_expr1), one, sub_expr2)], instance_expr = Equals(v, VecSum(index_or_indices = [i], summand = ScalarMult(lambda_i, TensorProd(a_i, b_i)), domain = Interval(one, sub_expr2))), domain = RealNonNeg).with_wrapping(), condition = InSet(Set(b_1_to_n), OrthoNormBases(B))).with_wrapping(), condition = InSet(Set(a_1_to_m), OrthoNormBases(A))).with_wrapping(), domain = TensorProd(A, B))

# 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_{v \in A {\otimes} B}~\left[\begin{array}{l}\exists_{a_{1}, a_{2}, \ldots, a_{m}~|~\left\{a_{1}, a_{2}, \ldots, a_{m}\right\} \in \textrm{O.N.Bases}\left(A\right)}~\\
\left[\begin{array}{l}\exists_{b_{1}, b_{2}, \ldots, b_{n}~|~\left\{b_{1}, b_{2}, \ldots, b_{n}\right\} \in \textrm{O.N.Bases}\left(B\right)}~\\
\left[\begin{array}{l}\exists_{\lambda_{1}, \lambda_{2}, \ldots, \lambda_{{\rm Min}\left(m, n\right)} \in \mathbb{R}^{\ge 0}}~\\
\left(v = \left(\sum_{i=1}^{{\rm Min}\left(m, n\right)} \left(\lambda_{i} \cdot \left(a_{i} {\otimes} b_{i}\right)\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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameter: 53 body: 5
4	ExprTuple	53
5	Conditional	value: 6 condition: 7
6	Operation	operator: 28 operand: 10
7	Operation	operator: 70 operands: 9
8	ExprTuple	10
9	ExprTuple	53, 11
10	Lambda	parameters: 22 body: 12
11	Operation	operator: 78 operands: 13
12	Conditional	value: 14 condition: 15
13	ExprTuple	27, 46
14	Operation	operator: 28 operand: 18
15	Operation	operator: 70 operands: 17
16	ExprTuple	18
17	ExprTuple	19, 20
18	Lambda	parameters: 39 body: 21
19	Operation	operator: 38 operands: 22
20	Operation	operator: 40 operand: 27
21	Conditional	value: 24 condition: 25
22	ExprTuple	26
23	ExprTuple	27
24	Operation	operator: 28 operand: 32
25	Operation	operator: 70 operands: 30
26	ExprRange	lambda_map: 31 start_index: 84 end_index: 92
27	Variable
28	Literal
29	ExprTuple	32
30	ExprTuple	33, 34
31	Lambda	parameter: 76 body: 35
32	Lambda	parameters: 36 body: 37
33	Operation	operator: 38 operands: 39
34	Operation	operator: 40 operand: 46
35	IndexedVar	variable: 86 index: 76
36	ExprTuple	42
37	Conditional	value: 43 condition: 44
38	Literal
39	ExprTuple	45
40	Literal
41	ExprTuple	46
42	ExprRange	lambda_map: 47 start_index: 84 end_index: 85
43	Operation	operator: 48 operands: 49
44	Operation	operator: 50 operands: 51
45	ExprRange	lambda_map: 52 start_index: 84 end_index: 93
46	Variable
47	Lambda	parameter: 76 body: 66
48	Literal
49	ExprTuple	53, 54
50	Literal
51	ExprTuple	55
52	Lambda	parameter: 76 body: 56
53	Variable
54	Operation	operator: 57 operand: 60
55	ExprRange	lambda_map: 59 start_index: 84 end_index: 85
56	IndexedVar	variable: 87 index: 76
57	Literal
58	ExprTuple	60
59	Lambda	parameter: 76 body: 61
60	Lambda	parameter: 91 body: 62
61	Operation	operator: 70 operands: 63
62	Conditional	value: 64 condition: 65
63	ExprTuple	66, 67
64	Operation	operator: 68 operands: 69
65	Operation	operator: 70 operands: 71
66	IndexedVar	variable: 77 index: 76
67	Literal
68	Literal
69	ExprTuple	73, 74
70	Literal
71	ExprTuple	91, 75
72	ExprTuple	76
73	IndexedVar	variable: 77 index: 91
74	Operation	operator: 78 operands: 79
75	Operation	operator: 80 operands: 81
76	Variable
77	Variable
78	Literal
79	ExprTuple	82, 83
80	Literal
81	ExprTuple	84, 85
82	IndexedVar	variable: 86 index: 91
83	IndexedVar	variable: 87 index: 91
84	Literal
85	Operation	operator: 89 operands: 90
86	Variable
87	Variable
88	ExprTuple	91
89	Literal
90	ExprTuple	92, 93
91	Variable
92	Variable
93	Variable

Expression of type Forall¶

from the theory of proveit.linear_algebra.inner_products¶

Expression of type Forall ¶

from the theory of proveit.linear_algebra.inner_products ¶