Expression of type Conditional ¶

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, f, i, n
from proveit.core_expr_types import a_1_to_n, a_i, 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 = 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)))

# 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\{\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()

# display the expression information
stored_expr.expr_info()

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

Expression of type Conditional¶

from the theory of proveit.physics.quantum.algebra¶

Expression of type Conditional ¶

from the theory of proveit.physics.quantum.algebra ¶