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 Conditional, Function, Lambda, f, i, n
from proveit.core_expr_types import a_1_to_n, a_i, v_1_to_n, v_i
from proveit.linear_algebra import Dim, HilbertSpaces, Hspace, OrthoNormBases, ScalarMult, VecSum
from proveit.logic import And, Equals, Forall, Functions, InClass, InSet, Set
from proveit.numbers import Complex, Interval, one
from proveit.physics.quantum import Bra, Ket, Qmult
from proveit.physics.quantum.algebra import v_1_to_n_kets

# build up the expression from sub-expressions
sub_expr1 = [i]
sub_expr2 = Interval(one, n)
sub_expr3 = Qmult(Ket(v_i), Bra(v_i))
expr = Lambda(Hspace, Conditional(Forall(instance_param_or_params = [v_1_to_n], instance_expr = Forall(instance_param_or_params = [a_1_to_n], instance_expr = Forall(instance_param_or_params = [f], instance_expr = Equals(Function(f, [VecSum(index_or_indices = sub_expr1, summand = ScalarMult(a_i, sub_expr3), domain = sub_expr2)]), VecSum(index_or_indices = sub_expr1, summand = ScalarMult(Function(f, [a_i]), sub_expr3), domain = sub_expr2)), domain = Functions(Complex, Complex)), domain = Complex), condition = InSet(Set(v_1_to_n_kets), OrthoNormBases(Hspace))).with_wrapping(), And(InClass(Hspace, HilbertSpaces), Equals(Dim(Hspace), n))))

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())

\mathcal{H} \mapsto \left\{\begin{array}{l}\forall_{v_{1}, v_{2}, \ldots, v_{n}~|~\left\{\left\{\lvert v_{1} \rangle, \lvert v_{2} \rangle, \ldots, \lvert v_{n} \rangle\right\}\right\} \in \textrm{O.N.Bases}\left(\mathcal{H}\right)}~\\
\left[\forall_{a_{1}, a_{2}, \ldots, a_{n} \in \mathbb{C}}~\left[\forall_{f \in \left[\mathbb{C} \rightarrow \mathbb{C}\right]}~\left(f\left(\sum_{i=1}^{n} \left(a_{i} \cdot \left(\lvert v_{i} \rangle \thinspace \langle v_{i} \rvert\right)\right)\right) = \left(\sum_{i=1}^{n} \left(f\left(a_{i}\right) \cdot \left(\lvert v_{i} \rangle \thinspace \langle v_{i} \rvert\right)\right)\right)\right)\right]\right]\end{array} \textrm{ if } \mathcal{H} \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces} ,  \textrm{Dim}\left(\mathcal{H}\right) = n\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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