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 Equals, Forall, Functions, InSet, Set
from proveit.numbers import Complex, Interval, NaturalPos, 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(n, Conditional(Forall(instance_param_or_params = [Hspace], instance_expr = 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(), domain = HilbertSpaces, condition = Equals(Dim(Hspace), n)).with_wrapping(), 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())

n \mapsto \left\{\begin{array}{l}\forall_{\mathcal{H} \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces}~|~\textrm{Dim}\left(\mathcal{H}\right) = n}~\\
\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}\right]\end{array} \textrm{ if } 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	parameter: 108 body: 2
1	ExprTuple	108
2	Conditional	value: 3 condition: 4
3	Operation	operator: 45 operand: 7
4	Operation	operator: 93 operands: 6
5	ExprTuple	7
6	ExprTuple	108, 8
7	Lambda	parameter: 43 body: 9
8	Literal
9	Conditional	value: 10 condition: 11
10	Operation	operator: 45 operand: 14
11	Operation	operator: 47 operands: 13
12	ExprTuple	14
13	ExprTuple	15, 16
14	Lambda	parameters: 17 body: 18
15	Operation	operator: 19 operands: 20
16	Operation	operator: 62 operands: 21
17	ExprTuple	22
18	Conditional	value: 23 condition: 24
19	Literal
20	ExprTuple	43, 25
21	ExprTuple	26, 108
22	ExprRange	lambda_map: 27 start_index: 107 end_index: 108
23	Operation	operator: 45 operand: 31
24	Operation	operator: 93 operands: 29
25	Literal
26	Operation	operator: 30 operand: 43
27	Lambda	parameter: 80 body: 71
28	ExprTuple	31
29	ExprTuple	32, 33
30	Literal
31	Lambda	parameters: 34 body: 35
32	Operation	operator: 49 operand: 42
33	Operation	operator: 37 operand: 43
34	ExprTuple	39
35	Conditional	value: 40 condition: 41
36	ExprTuple	42
37	Literal
38	ExprTuple	43
39	ExprRange	lambda_map: 44 start_index: 107 end_index: 108
40	Operation	operator: 45 operand: 51
41	Operation	operator: 47 operands: 48
42	Operation	operator: 49 operands: 50
43	Variable
44	Lambda	parameter: 80 body: 70
45	Literal
46	ExprTuple	51
47	Literal
48	ExprTuple	52
49	Literal
50	ExprTuple	53
51	Lambda	parameter: 95 body: 55
52	ExprRange	lambda_map: 56 start_index: 107 end_index: 108
53	ExprRange	lambda_map: 57 start_index: 107 end_index: 108
54	ExprTuple	95
55	Conditional	value: 58 condition: 59
56	Lambda	parameter: 80 body: 60
57	Lambda	parameter: 80 body: 61
58	Operation	operator: 62 operands: 63
59	Operation	operator: 93 operands: 64
60	Operation	operator: 93 operands: 65
61	Operation	operator: 109 operand: 71
62	Literal
63	ExprTuple	67, 68
64	ExprTuple	95, 69
65	ExprTuple	70, 79
66	ExprTuple	71
67	Operation	operator: 95 operand: 77
68	Operation	operator: 81 operand: 78
69	Operation	operator: 74 operands: 75
70	IndexedVar	variable: 104 index: 80
71	IndexedVar	variable: 113 index: 80
72	ExprTuple	77
73	ExprTuple	78
74	Literal
75	ExprTuple	79, 79
76	ExprTuple	80
77	Operation	operator: 81 operand: 84
78	Lambda	parameter: 115 body: 83
79	Literal
80	Variable
81	Literal
82	ExprTuple	84
83	Conditional	value: 85 condition: 89
84	Lambda	parameter: 115 body: 86
85	Operation	operator: 91 operands: 87
86	Conditional	value: 88 condition: 89
87	ExprTuple	90, 97
88	Operation	operator: 91 operands: 92
89	Operation	operator: 93 operands: 94
90	Operation	operator: 95 operand: 99
91	Literal
92	ExprTuple	99, 97
93	Literal
94	ExprTuple	115, 98
95	Variable
96	ExprTuple	99
97	Operation	operator: 100 operands: 101
98	Operation	operator: 102 operands: 103
99	IndexedVar	variable: 104 index: 115
100	Literal
101	ExprTuple	105, 106
102	Literal
103	ExprTuple	107, 108
104	Variable
105	Operation	operator: 109 operand: 112
106	Operation	operator: 110 operand: 112
107	Literal
108	Variable
109	Literal
110	Literal
111	ExprTuple	112
112	IndexedVar	variable: 113 index: 115
113	Variable
114	ExprTuple	115
115	Variable