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 LinMap, OrthoNormBases, VecSum
from proveit.logic import 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(A, Conditional(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), InSet(A, LinMap(V, W))))

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())

A \mapsto \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] \textrm{ if } A \in \mathcal{L}\left(V, W\right)\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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