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 Dim, HilbertSpaces, 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 = [A, B], instance_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)), domain = HilbertSpaces, conditions = [Equals(Dim(A), m), Equals(Dim(B), n)]).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_{A, B \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces}~|~\textrm{Dim}\left(A\right) = m, \textrm{Dim}\left(B\right) = n}~\\
\left[\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]\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: 6 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 32 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 6 operand: 9
5	Operation	operator: 69 operands: 8
6	Literal
7	ExprTuple	9
8	ExprTuple	10, 11, 12, 13
9	Lambda	parameter: 72 body: 15
10	Operation	operator: 17 operands: 16
11	Operation	operator: 17 operands: 18
12	Operation	operator: 67 operands: 19
13	Operation	operator: 67 operands: 20
14	ExprTuple	72
15	Conditional	value: 21 condition: 22
16	ExprTuple	46, 23
17	Literal
18	ExprTuple	65, 23
19	ExprTuple	24, 111
20	ExprTuple	25, 112
21	Operation	operator: 47 operand: 29
22	Operation	operator: 89 operands: 27
23	Literal
24	Operation	operator: 28 operand: 46
25	Operation	operator: 28 operand: 65
26	ExprTuple	29
27	ExprTuple	72, 30
28	Literal
29	Lambda	parameters: 41 body: 31
30	Operation	operator: 97 operands: 32
31	Conditional	value: 33 condition: 34
32	ExprTuple	46, 65
33	Operation	operator: 47 operand: 37
34	Operation	operator: 89 operands: 36
35	ExprTuple	37
36	ExprTuple	38, 39
37	Lambda	parameters: 58 body: 40
38	Operation	operator: 57 operands: 41
39	Operation	operator: 59 operand: 46
40	Conditional	value: 43 condition: 44
41	ExprTuple	45
42	ExprTuple	46
43	Operation	operator: 47 operand: 51
44	Operation	operator: 89 operands: 49
45	ExprRange	lambda_map: 50 start_index: 103 end_index: 111
46	Variable
47	Literal
48	ExprTuple	51
49	ExprTuple	52, 53
50	Lambda	parameter: 95 body: 54
51	Lambda	parameters: 55 body: 56
52	Operation	operator: 57 operands: 58
53	Operation	operator: 59 operand: 65
54	IndexedVar	variable: 105 index: 95
55	ExprTuple	61
56	Conditional	value: 62 condition: 63
57	Literal
58	ExprTuple	64
59	Literal
60	ExprTuple	65
61	ExprRange	lambda_map: 66 start_index: 103 end_index: 104
62	Operation	operator: 67 operands: 68
63	Operation	operator: 69 operands: 70
64	ExprRange	lambda_map: 71 start_index: 103 end_index: 112
65	Variable
66	Lambda	parameter: 95 body: 85
67	Literal
68	ExprTuple	72, 73
69	Literal
70	ExprTuple	74
71	Lambda	parameter: 95 body: 75
72	Variable
73	Operation	operator: 76 operand: 79
74	ExprRange	lambda_map: 78 start_index: 103 end_index: 104
75	IndexedVar	variable: 106 index: 95
76	Literal
77	ExprTuple	79
78	Lambda	parameter: 95 body: 80
79	Lambda	parameter: 110 body: 81
80	Operation	operator: 89 operands: 82
81	Conditional	value: 83 condition: 84
82	ExprTuple	85, 86
83	Operation	operator: 87 operands: 88
84	Operation	operator: 89 operands: 90
85	IndexedVar	variable: 96 index: 95
86	Literal
87	Literal
88	ExprTuple	92, 93
89	Literal
90	ExprTuple	110, 94
91	ExprTuple	95
92	IndexedVar	variable: 96 index: 110
93	Operation	operator: 97 operands: 98
94	Operation	operator: 99 operands: 100
95	Variable
96	Variable
97	Literal
98	ExprTuple	101, 102
99	Literal
100	ExprTuple	103, 104
101	IndexedVar	variable: 105 index: 110
102	IndexedVar	variable: 106 index: 110
103	Literal
104	Operation	operator: 108 operands: 109
105	Variable
106	Variable
107	ExprTuple	110
108	Literal
109	ExprTuple	111, 112
110	Variable
111	Variable
112	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 ¶