Expression of type Forall ¶

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 Function, 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 = 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()

# 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())

\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}

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	True	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

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

Expression of type Forall¶

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

Expression of type Forall ¶

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