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 OrthoNormBases, VecSum
from proveit.logic import And, Equals, Forall, 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([m, n], Conditional(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(), 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_{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} \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: 1 body: 2
1	ExprTuple	71, 92
2	Conditional	value: 3 condition: 4
3	Operation	operator: 20 operand: 8
4	Operation	operator: 6 operands: 7
5	ExprTuple	8
6	Literal
7	ExprTuple	9, 10
8	Lambda	parameters: 11 body: 12
9	Operation	operator: 75 operands: 13
10	Operation	operator: 75 operands: 14
11	ExprTuple	15
12	Conditional	value: 16 condition: 17
13	ExprTuple	71, 18
14	ExprTuple	92, 18
15	ExprRange	lambda_map: 19 start_index: 91 end_index: 71
16	Operation	operator: 20 operand: 23
17	Operation	operator: 75 operands: 22
18	Literal
19	Lambda	parameter: 72 body: 53
20	Literal
21	ExprTuple	23
22	ExprTuple	24, 25
23	Lambda	parameters: 26 body: 27
24	Operation	operator: 45 operands: 28
25	Operation	operator: 47 operand: 34
26	ExprTuple	30
27	Conditional	value: 31 condition: 32
28	ExprTuple	33
29	ExprTuple	34
30	ExprRange	lambda_map: 35 start_index: 91 end_index: 92
31	Operation	operator: 36 operands: 37
32	Operation	operator: 75 operands: 38
33	ExprRange	lambda_map: 39 start_index: 91 end_index: 71
34	Variable
35	Lambda	parameter: 72 body: 65
36	Literal
37	ExprTuple	78, 40
38	ExprTuple	41, 42
39	Lambda	parameter: 72 body: 43
40	Operation	operator: 59 operand: 50
41	Operation	operator: 45 operands: 46
42	Operation	operator: 47 operand: 52
43	Operation	operator: 83 operand: 53
44	ExprTuple	50
45	Literal
46	ExprTuple	51
47	Literal
48	ExprTuple	52
49	ExprTuple	53
50	Lambda	parameter: 98 body: 54
51	ExprRange	lambda_map: 55 start_index: 91 end_index: 92
52	Variable
53	IndexedVar	variable: 95 index: 72
54	Conditional	value: 56 condition: 57
55	Lambda	parameter: 72 body: 58
56	Operation	operator: 59 operand: 63
57	Operation	operator: 75 operands: 61
58	Operation	operator: 83 operand: 65
59	Literal
60	ExprTuple	63
61	ExprTuple	98, 64
62	ExprTuple	65
63	Lambda	parameter: 97 body: 66
64	Operation	operator: 87 operands: 67
65	IndexedVar	variable: 93 index: 72
66	Conditional	value: 69 condition: 70
67	ExprTuple	91, 71
68	ExprTuple	72
69	Operation	operator: 73 operands: 74
70	Operation	operator: 75 operands: 76
71	Variable
72	Variable
73	Literal
74	ExprTuple	77, 78, 79, 80, 81
75	Literal
76	ExprTuple	97, 82
77	Operation	operator: 85 operand: 89
78	Variable
79	Operation	operator: 83 operand: 90
80	Operation	operator: 83 operand: 89
81	Operation	operator: 85 operand: 90
82	Operation	operator: 87 operands: 88
83	Literal
84	ExprTuple	89
85	Literal
86	ExprTuple	90
87	Literal
88	ExprTuple	91, 92
89	IndexedVar	variable: 93 index: 97
90	IndexedVar	variable: 95 index: 98
91	Literal
92	Variable
93	Variable
94	ExprTuple	97
95	Variable
96	ExprTuple	98
97	Variable
98	Variable