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 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 = Forall(instance_param_or_params = [n], instance_expr = 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(), domain = NaturalPos)

# 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())

\forall_{n \in \mathbb{N}^+}~\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}\right]

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	None/False	('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: 47 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 110 body: 4
3	ExprTuple	110
4	Conditional	value: 5 condition: 6
5	Operation	operator: 47 operand: 9
6	Operation	operator: 95 operands: 8
7	ExprTuple	9
8	ExprTuple	110, 10
9	Lambda	parameter: 45 body: 11
10	Literal
11	Conditional	value: 12 condition: 13
12	Operation	operator: 47 operand: 16
13	Operation	operator: 49 operands: 15
14	ExprTuple	16
15	ExprTuple	17, 18
16	Lambda	parameters: 19 body: 20
17	Operation	operator: 21 operands: 22
18	Operation	operator: 64 operands: 23
19	ExprTuple	24
20	Conditional	value: 25 condition: 26
21	Literal
22	ExprTuple	45, 27
23	ExprTuple	28, 110
24	ExprRange	lambda_map: 29 start_index: 109 end_index: 110
25	Operation	operator: 47 operand: 33
26	Operation	operator: 95 operands: 31
27	Literal
28	Operation	operator: 32 operand: 45
29	Lambda	parameter: 82 body: 73
30	ExprTuple	33
31	ExprTuple	34, 35
32	Literal
33	Lambda	parameters: 36 body: 37
34	Operation	operator: 51 operand: 44
35	Operation	operator: 39 operand: 45
36	ExprTuple	41
37	Conditional	value: 42 condition: 43
38	ExprTuple	44
39	Literal
40	ExprTuple	45
41	ExprRange	lambda_map: 46 start_index: 109 end_index: 110
42	Operation	operator: 47 operand: 53
43	Operation	operator: 49 operands: 50
44	Operation	operator: 51 operands: 52
45	Variable
46	Lambda	parameter: 82 body: 72
47	Literal
48	ExprTuple	53
49	Literal
50	ExprTuple	54
51	Literal
52	ExprTuple	55
53	Lambda	parameter: 97 body: 57
54	ExprRange	lambda_map: 58 start_index: 109 end_index: 110
55	ExprRange	lambda_map: 59 start_index: 109 end_index: 110
56	ExprTuple	97
57	Conditional	value: 60 condition: 61
58	Lambda	parameter: 82 body: 62
59	Lambda	parameter: 82 body: 63
60	Operation	operator: 64 operands: 65
61	Operation	operator: 95 operands: 66
62	Operation	operator: 95 operands: 67
63	Operation	operator: 111 operand: 73
64	Literal
65	ExprTuple	69, 70
66	ExprTuple	97, 71
67	ExprTuple	72, 81
68	ExprTuple	73
69	Operation	operator: 97 operand: 79
70	Operation	operator: 83 operand: 80
71	Operation	operator: 76 operands: 77
72	IndexedVar	variable: 106 index: 82
73	IndexedVar	variable: 115 index: 82
74	ExprTuple	79
75	ExprTuple	80
76	Literal
77	ExprTuple	81, 81
78	ExprTuple	82
79	Operation	operator: 83 operand: 86
80	Lambda	parameter: 117 body: 85
81	Literal
82	Variable
83	Literal
84	ExprTuple	86
85	Conditional	value: 87 condition: 91
86	Lambda	parameter: 117 body: 88
87	Operation	operator: 93 operands: 89
88	Conditional	value: 90 condition: 91
89	ExprTuple	92, 99
90	Operation	operator: 93 operands: 94
91	Operation	operator: 95 operands: 96
92	Operation	operator: 97 operand: 101
93	Literal
94	ExprTuple	101, 99
95	Literal
96	ExprTuple	117, 100
97	Variable
98	ExprTuple	101
99	Operation	operator: 102 operands: 103
100	Operation	operator: 104 operands: 105
101	IndexedVar	variable: 106 index: 117
102	Literal
103	ExprTuple	107, 108
104	Literal
105	ExprTuple	109, 110
106	Variable
107	Operation	operator: 111 operand: 114
108	Operation	operator: 112 operand: 114
109	Literal
110	Variable
111	Literal
112	Literal
113	ExprTuple	114
114	IndexedVar	variable: 115 index: 117
115	Variable
116	ExprTuple	117
117	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 ¶