Theorem schmidt_decomposition of type Forall ¶

from the theory of proveit.linear_algebra.inner_products ¶

In [1]:

import proveit
# Automation is not needed when only building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_theorem_expr # Load the stored theorem expression as 'stored_expr'
# import the special expression
from proveit.linear_algebra.inner_products import schmidt_decomposition

In [2]:

# check that the built expression is the same as the stored expression
assert schmidt_decomposition.expr == stored_expr
assert schmidt_decomposition.expr._style_id == stored_expr._style_id
print("Passed sanity check: schmidt_decomposition matches stored_expr")

Passed sanity check: schmidt_decomposition matches stored_expr

In [3]:

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\forall_{m, n \in \mathbb{N}^+}~\left[\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}\right]

In [4]:

stored_expr.style_options()

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',)

In [5]:

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions
0	Operation	operator: 17 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 120 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 17 operand: 8
5	Operation	operator: 80 operands: 7
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameters: 43 body: 11
9	Operation	operator: 100 operands: 12
10	Operation	operator: 100 operands: 13
11	Conditional	value: 14 condition: 15
12	ExprTuple	122, 16
13	ExprTuple	123, 16
14	Operation	operator: 17 operand: 20
15	Operation	operator: 80 operands: 19
16	Literal
17	Literal
18	ExprTuple	20
19	ExprTuple	21, 22, 23, 24
20	Lambda	parameter: 83 body: 26
21	Operation	operator: 28 operands: 27
22	Operation	operator: 28 operands: 29
23	Operation	operator: 78 operands: 30
24	Operation	operator: 78 operands: 31
25	ExprTuple	83
26	Conditional	value: 32 condition: 33
27	ExprTuple	57, 34
28	Literal
29	ExprTuple	76, 34
30	ExprTuple	35, 122
31	ExprTuple	36, 123
32	Operation	operator: 58 operand: 40
33	Operation	operator: 100 operands: 38
34	Literal
35	Operation	operator: 39 operand: 57
36	Operation	operator: 39 operand: 76
37	ExprTuple	40
38	ExprTuple	83, 41
39	Literal
40	Lambda	parameters: 52 body: 42
41	Operation	operator: 108 operands: 43
42	Conditional	value: 44 condition: 45
43	ExprTuple	57, 76
44	Operation	operator: 58 operand: 48
45	Operation	operator: 100 operands: 47
46	ExprTuple	48
47	ExprTuple	49, 50
48	Lambda	parameters: 69 body: 51
49	Operation	operator: 68 operands: 52
50	Operation	operator: 70 operand: 57
51	Conditional	value: 54 condition: 55
52	ExprTuple	56
53	ExprTuple	57
54	Operation	operator: 58 operand: 62
55	Operation	operator: 100 operands: 60
56	ExprRange	lambda_map: 61 start_index: 114 end_index: 122
57	Variable
58	Literal
59	ExprTuple	62
60	ExprTuple	63, 64
61	Lambda	parameter: 106 body: 65
62	Lambda	parameters: 66 body: 67
63	Operation	operator: 68 operands: 69
64	Operation	operator: 70 operand: 76
65	IndexedVar	variable: 116 index: 106
66	ExprTuple	72
67	Conditional	value: 73 condition: 74
68	Literal
69	ExprTuple	75
70	Literal
71	ExprTuple	76
72	ExprRange	lambda_map: 77 start_index: 114 end_index: 115
73	Operation	operator: 78 operands: 79
74	Operation	operator: 80 operands: 81
75	ExprRange	lambda_map: 82 start_index: 114 end_index: 123
76	Variable
77	Lambda	parameter: 106 body: 96
78	Literal
79	ExprTuple	83, 84
80	Literal
81	ExprTuple	85
82	Lambda	parameter: 106 body: 86
83	Variable
84	Operation	operator: 87 operand: 90
85	ExprRange	lambda_map: 89 start_index: 114 end_index: 115
86	IndexedVar	variable: 117 index: 106
87	Literal
88	ExprTuple	90
89	Lambda	parameter: 106 body: 91
90	Lambda	parameter: 121 body: 92
91	Operation	operator: 100 operands: 93
92	Conditional	value: 94 condition: 95
93	ExprTuple	96, 97
94	Operation	operator: 98 operands: 99
95	Operation	operator: 100 operands: 101
96	IndexedVar	variable: 107 index: 106
97	Literal
98	Literal
99	ExprTuple	103, 104
100	Literal
101	ExprTuple	121, 105
102	ExprTuple	106
103	IndexedVar	variable: 107 index: 121
104	Operation	operator: 108 operands: 109
105	Operation	operator: 110 operands: 111
106	Variable
107	Variable
108	Literal
109	ExprTuple	112, 113
110	Literal
111	ExprTuple	114, 115
112	IndexedVar	variable: 116 index: 121
113	IndexedVar	variable: 117 index: 121
114	Literal
115	Operation	operator: 119 operands: 120
116	Variable
117	Variable
118	ExprTuple	121
119	Literal
120	ExprTuple	122, 123
121	Variable
122	Variable
123	Variable