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 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 = 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(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..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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