Expression of type Lambda

from the theory of proveit.physics.quantum.algebra

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, Conditional, Lambda, V, W, i, j, m, n
from proveit.core_expr_types import v_1_to_m, v_i, w_1_to_n, w_j
from proveit.linear_algebra import HilbertSpaces, LinMap, OrthoNormBases, VecSum
from proveit.logic import And, Equals, Forall, InClass, InSet
from proveit.numbers import Interval, NaturalPos, one
from proveit.physics.quantum import Bra, Ket, Qmult
from proveit.physics.quantum.algebra import v_1_to_m_kets, w_1_to_n_kets

# build up the expression from sub-expressions
expr = Lambda([V, W], Conditional(Forall(instance_param_or_params = [A], instance_expr = Forall(instance_param_or_params = [m, n], instance_expr = Forall(instance_param_or_params = [v_1_to_m], instance_expr = Forall(instance_param_or_params = [w_1_to_n], instance_expr = Equals(A, VecSum(index_or_indices = [i], summand = VecSum(index_or_indices = [j], summand = Qmult(Bra(w_j), A, Ket(v_i), Ket(w_j), Bra(v_i)), domain = Interval(one, n)), domain = Interval(one, m))), condition = InSet(w_1_to_n_kets, OrthoNormBases(W))).with_wrapping(), condition = InSet(v_1_to_m_kets, OrthoNormBases(V))).with_wrapping(), domain = NaturalPos), domain = LinMap(V, W)), And(InClass(V, HilbertSpaces), InClass(W, HilbertSpaces))))

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(V, W\right) \mapsto \left\{\forall_{A \in \mathcal{L}\left(V, W\right)}~\left[\forall_{m, n \in \mathbb{N}^+}~\left[\begin{array}{l}\forall_{v_{1}, v_{2}, \ldots, v_{m}~|~\left\{\lvert v_{1} \rangle, \lvert v_{2} \rangle, \ldots, \lvert v_{m} \rangle\right\} \in \textrm{O.N.Bases}\left(V\right)}~\\
\left[\begin{array}{l}\forall_{w_{1}, w_{2}, \ldots, w_{n}~|~\left\{\lvert w_{1} \rangle, \lvert w_{2} \rangle, \ldots, \lvert w_{n} \rangle\right\} \in \textrm{O.N.Bases}\left(W\right)}~\\
\left(A = \left(\sum_{i=1}^{m} \left(\sum_{j=1}^{n} \left(\langle w_{j} \rvert \thinspace A \thinspace \lvert v_{i} \rangle \thinspace \lvert w_{j} \rangle \thinspace \langle v_{i} \rvert\right)\right)\right)\right)\end{array}\right]\end{array}\right]\right] \textrm{ if } V \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces} ,  W \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces}\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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