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 Hspace, OrthoNormBases, ScalarMult, VecSum
from proveit.logic import Equals, Forall, Functions, 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 = ExprTuple(Lambda([v_1_to_n], Conditional(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), InSet(Set(v_1_to_n_kets), OrthoNormBases(Hspace)))))

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