Expression of type ExprTuple

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, ExprTuple, 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 = ExprTuple(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())

\left(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..\right)

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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