Expression of type Lambda

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, Conditional, ExprRange, IndexedVar, Lambda, 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 And, Equals, Exists, Forall, InSet, Set
from proveit.numbers import Interval, Min, NaturalPos, RealNonNeg, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Min(m, n)
expr = Lambda([m, n], Conditional(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(), And(InSet(m, NaturalPos), InSet(n, NaturalPos))))

expr:

# 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())

\left(m, n\right) \mapsto \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} \textrm{ if } m \in \mathbb{N}^+ ,  n \in \mathbb{N}^+\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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