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, 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 = [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()

# 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_{\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}

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: 39 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 37 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 39 operand: 8
5	Operation	operator: 41 operands: 7
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameters: 11 body: 12
9	Operation	operator: 13 operands: 14
10	Operation	operator: 56 operands: 15
11	ExprTuple	16
12	Conditional	value: 17 condition: 18
13	Literal
14	ExprTuple	37, 19
15	ExprTuple	20, 102
16	ExprRange	lambda_map: 21 start_index: 101 end_index: 102
17	Operation	operator: 39 operand: 25
18	Operation	operator: 87 operands: 23
19	Literal
20	Operation	operator: 24 operand: 37
21	Lambda	parameter: 74 body: 65
22	ExprTuple	25
23	ExprTuple	26, 27
24	Literal
25	Lambda	parameters: 28 body: 29
26	Operation	operator: 43 operand: 36
27	Operation	operator: 31 operand: 37
28	ExprTuple	33
29	Conditional	value: 34 condition: 35
30	ExprTuple	36
31	Literal
32	ExprTuple	37
33	ExprRange	lambda_map: 38 start_index: 101 end_index: 102
34	Operation	operator: 39 operand: 45
35	Operation	operator: 41 operands: 42
36	Operation	operator: 43 operands: 44
37	Variable
38	Lambda	parameter: 74 body: 64
39	Literal
40	ExprTuple	45
41	Literal
42	ExprTuple	46
43	Literal
44	ExprTuple	47
45	Lambda	parameter: 89 body: 49
46	ExprRange	lambda_map: 50 start_index: 101 end_index: 102
47	ExprRange	lambda_map: 51 start_index: 101 end_index: 102
48	ExprTuple	89
49	Conditional	value: 52 condition: 53
50	Lambda	parameter: 74 body: 54
51	Lambda	parameter: 74 body: 55
52	Operation	operator: 56 operands: 57
53	Operation	operator: 87 operands: 58
54	Operation	operator: 87 operands: 59
55	Operation	operator: 103 operand: 65
56	Literal
57	ExprTuple	61, 62
58	ExprTuple	89, 63
59	ExprTuple	64, 73
60	ExprTuple	65
61	Operation	operator: 89 operand: 71
62	Operation	operator: 75 operand: 72
63	Operation	operator: 68 operands: 69
64	IndexedVar	variable: 98 index: 74
65	IndexedVar	variable: 107 index: 74
66	ExprTuple	71
67	ExprTuple	72
68	Literal
69	ExprTuple	73, 73
70	ExprTuple	74
71	Operation	operator: 75 operand: 78
72	Lambda	parameter: 109 body: 77
73	Literal
74	Variable
75	Literal
76	ExprTuple	78
77	Conditional	value: 79 condition: 83
78	Lambda	parameter: 109 body: 80
79	Operation	operator: 85 operands: 81
80	Conditional	value: 82 condition: 83
81	ExprTuple	84, 91
82	Operation	operator: 85 operands: 86
83	Operation	operator: 87 operands: 88
84	Operation	operator: 89 operand: 93
85	Literal
86	ExprTuple	93, 91
87	Literal
88	ExprTuple	109, 92
89	Variable
90	ExprTuple	93
91	Operation	operator: 94 operands: 95
92	Operation	operator: 96 operands: 97
93	IndexedVar	variable: 98 index: 109
94	Literal
95	ExprTuple	99, 100
96	Literal
97	ExprTuple	101, 102
98	Variable
99	Operation	operator: 103 operand: 106
100	Operation	operator: 104 operand: 106
101	Literal
102	Variable
103	Literal
104	Literal
105	ExprTuple	106
106	IndexedVar	variable: 107 index: 109
107	Variable
108	ExprTuple	109
109	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 ¶