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 M, 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 Adj, Dim, HilbertSpaces, Hspace, LinMap, OrthoNormBases, ScalarMult, VecSum
from proveit.logic import Equals, Exists, Forall, Iff, 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 = Adj(M)
expr = Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [Hspace], instance_expr = Forall(instance_param_or_params = [M], instance_expr = Iff(Equals(Qmult(sub_expr1, M), Qmult(M, sub_expr1)), Exists(instance_param_or_params = [v_1_to_n], instance_expr = Exists(instance_param_or_params = [a_1_to_n], instance_expr = Equals(M, VecSum(index_or_indices = [i], summand = ScalarMult(a_i, Qmult(Ket(v_i), Bra(v_i))), domain = Interval(one, n))), domain = Complex).with_wrapping(), condition = InSet(Set(v_1_to_n_kets), OrthoNormBases(Hspace))).with_wrapping()).with_wrapping_at(2), domain = LinMap(Hspace, Hspace)), 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[\forall_{M \in \mathcal{L}\left(\mathcal{H}, \mathcal{H}\right)}~\left(\begin{array}{c} \begin{array}{l} \left(\left(M^{\dagger} \thinspace M\right) = \left(M \thinspace M^{\dagger}\right)\right) \Leftrightarrow  \\ \left[\begin{array}{l}\exists_{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[\begin{array}{l}\exists_{a_{1}, a_{2}, \ldots, a_{n} \in \mathbb{C}}~\\
\left(M = \left(\sum_{i=1}^{n} \left(a_{i} \cdot \left(\lvert v_{i} \rangle \thinspace \langle v_{i} \rvert\right)\right)\right)\right)\end{array}\right]\end{array}\right] \end{array} \end{array}\right)\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: 14 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 112 body: 4
3	ExprTuple	112
4	Conditional	value: 5 condition: 6
5	Operation	operator: 14 operand: 9
6	Operation	operator: 97 operands: 8
7	ExprTuple	9
8	ExprTuple	112, 10
9	Lambda	parameter: 68 body: 11
10	Literal
11	Conditional	value: 12 condition: 13
12	Operation	operator: 14 operand: 17
13	Operation	operator: 72 operands: 16
14	Literal
15	ExprTuple	17
16	ExprTuple	18, 19
17	Lambda	parameter: 76 body: 20
18	Operation	operator: 21 operands: 22
19	Operation	operator: 70 operands: 23
20	Conditional	value: 24 condition: 25
21	Literal
22	ExprTuple	68, 26
23	ExprTuple	27, 112
24	Operation	operator: 28 operands: 29
25	Operation	operator: 97 operands: 30
26	Literal
27	Operation	operator: 31 operand: 68
28	Literal
29	ExprTuple	32, 33
30	ExprTuple	76, 34
31	Literal
32	Operation	operator: 70 operands: 35
33	Operation	operator: 53 operand: 41
34	Operation	operator: 37 operands: 38
35	ExprTuple	39, 40
36	ExprTuple	41
37	Literal
38	ExprTuple	68, 68
39	Operation	operator: 105 operands: 42
40	Operation	operator: 105 operands: 43
41	Lambda	parameters: 44 body: 45
42	ExprTuple	46, 76
43	ExprTuple	76, 46
44	ExprTuple	47
45	Conditional	value: 48 condition: 49
46	Operation	operator: 50 operand: 76
47	ExprRange	lambda_map: 52 start_index: 111 end_index: 112
48	Operation	operator: 53 operand: 56
49	Operation	operator: 97 operands: 55
50	Literal
51	ExprTuple	76
52	Lambda	parameter: 103 body: 94
53	Literal
54	ExprTuple	56
55	ExprTuple	57, 58
56	Lambda	parameters: 59 body: 60
57	Operation	operator: 74 operand: 67
58	Operation	operator: 62 operand: 68
59	ExprTuple	64
60	Conditional	value: 65 condition: 66
61	ExprTuple	67
62	Literal
63	ExprTuple	68
64	ExprRange	lambda_map: 69 start_index: 111 end_index: 112
65	Operation	operator: 70 operands: 71
66	Operation	operator: 72 operands: 73
67	Operation	operator: 74 operands: 75
68	Variable
69	Lambda	parameter: 103 body: 92
70	Literal
71	ExprTuple	76, 77
72	Literal
73	ExprTuple	78
74	Literal
75	ExprTuple	79
76	Variable
77	Operation	operator: 80 operand: 84
78	ExprRange	lambda_map: 82 start_index: 111 end_index: 112
79	ExprRange	lambda_map: 83 start_index: 111 end_index: 112
80	Literal
81	ExprTuple	84
82	Lambda	parameter: 103 body: 85
83	Lambda	parameter: 103 body: 86
84	Lambda	parameter: 119 body: 87
85	Operation	operator: 97 operands: 88
86	Operation	operator: 113 operand: 94
87	Conditional	value: 90 condition: 91
88	ExprTuple	92, 93
89	ExprTuple	94
90	Operation	operator: 95 operands: 96
91	Operation	operator: 97 operands: 98
92	IndexedVar	variable: 104 index: 103
93	Literal
94	IndexedVar	variable: 117 index: 103
95	Literal
96	ExprTuple	100, 101
97	Literal
98	ExprTuple	119, 102
99	ExprTuple	103
100	IndexedVar	variable: 104 index: 119
101	Operation	operator: 105 operands: 106
102	Operation	operator: 107 operands: 108
103	Variable
104	Variable
105	Literal
106	ExprTuple	109, 110
107	Literal
108	ExprTuple	111, 112
109	Operation	operator: 113 operand: 116
110	Operation	operator: 114 operand: 116
111	Literal
112	Variable
113	Literal
114	Literal
115	ExprTuple	116
116	IndexedVar	variable: 117 index: 119
117	Variable
118	ExprTuple	119
119	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 ¶