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 A, B, i, n
from proveit.core_expr_types import a_1_to_n, a_i, b_1_to_n, b_i, v_1_to_n, v_i
from proveit.linear_algebra import Adj, Commutator, MatrixSpace, OrthoNormBases, ScalarMult, VecSum, VecZero
from proveit.logic import And, CartExp, Equals, Exists, Forall, Iff, InSet, Set
from proveit.numbers import Complex, Interval, NaturalPos, one
from proveit.physics.quantum import Bra, Ket, Qmult

# build up the expression from sub-expressions
sub_expr1 = [i]
sub_expr2 = MatrixSpace(field = Complex, rows = n, columns = n)
sub_expr3 = Interval(one, n)
sub_expr4 = Qmult(Ket(v_i), Bra(v_i))
expr = Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [A, B], instance_expr = Iff(Equals(Commutator(A, B), VecZero(sub_expr2)), Exists(instance_param_or_params = [v_1_to_n], instance_expr = Exists(instance_param_or_params = [a_1_to_n, b_1_to_n], instance_expr = And(Equals(A, VecSum(index_or_indices = sub_expr1, summand = ScalarMult(a_i, sub_expr4), domain = sub_expr3)), Equals(B, VecSum(index_or_indices = sub_expr1, summand = ScalarMult(b_i, sub_expr4), domain = sub_expr3))).with_wrapping_at(2), domain = Complex).with_wrapping(), condition = InSet(Set(v_1_to_n), OrthoNormBases(CartExp(Complex, n)))).with_wrapping()).with_wrapping_at(2), domain = sub_expr2, conditions = [Equals(Adj(A), A), Equals(Adj(B), B)]).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_{A, B \in \mathbb{C}^{n \times n}~|~A^{\dagger} = A, B^{\dagger} = B}~\\
\left(\begin{array}{c} \begin{array}{l} \left(\left[A, B\right] = \vec{0}\left(\mathbb{C}^{n \times n}\right)\right) \Leftrightarrow  \\ \left[\begin{array}{l}\exists_{v_{1}, v_{2}, \ldots, v_{n}~|~\left\{v_{1}, v_{2}, \ldots, v_{n}\right\} \in \textrm{O.N.Bases}\left(\mathbb{C}^{n}\right)}~\\
\left[\begin{array}{l}\exists_{a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n} \in \mathbb{C}}~\\
\left(\begin{array}{c} \left(A = \left(\sum_{i=1}^{n} \left(a_{i} \cdot \left(\lvert v_{i} \rangle \thinspace \langle v_{i} \rvert\right)\right)\right)\right) \land  \\ \left(B = \left(\sum_{i=1}^{n} \left(b_{i} \cdot \left(\lvert v_{i} \rangle \thinspace \langle v_{i} \rvert\right)\right)\right)\right) \end{array}\right)\end{array}\right]\end{array}\right] \end{array} \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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 125 body: 4
3	ExprTuple	125
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 110 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	125, 11
10	Lambda	parameters: 36 body: 12
11	Literal
12	Conditional	value: 13 condition: 14
13	Operation	operator: 15 operands: 16
14	Operation	operator: 69 operands: 17
15	Literal
16	ExprTuple	18, 19
17	ExprTuple	20, 21, 22, 23
18	Operation	operator: 80 operands: 24
19	Operation	operator: 48 operand: 32
20	Operation	operator: 110 operands: 26
21	Operation	operator: 110 operands: 27
22	Operation	operator: 80 operands: 28
23	Operation	operator: 80 operands: 29
24	ExprTuple	30, 31
25	ExprTuple	32
26	ExprTuple	84, 43
27	ExprTuple	86, 43
28	ExprTuple	33, 84
29	ExprTuple	34, 86
30	Operation	operator: 35 operands: 36
31	Operation	operator: 37 operand: 43
32	Lambda	parameters: 57 body: 39
33	Operation	operator: 41 operand: 84
34	Operation	operator: 41 operand: 86
35	Literal
36	ExprTuple	84, 86
37	Literal
38	ExprTuple	43
39	Conditional	value: 44 condition: 45
40	ExprTuple	84
41	Literal
42	ExprTuple	86
43	Operation	operator: 46 operands: 47
44	Operation	operator: 48 operand: 51
45	Operation	operator: 110 operands: 50
46	Literal
47	NamedExprs	field: 99 rows: 125 columns: 125
48	Literal
49	ExprTuple	51
50	ExprTuple	52, 53
51	Lambda	parameters: 54 body: 55
52	Operation	operator: 56 operands: 57
53	Operation	operator: 58 operand: 65
54	ExprTuple	60, 61
55	Conditional	value: 62 condition: 63
56	Literal
57	ExprTuple	64
58	Literal
59	ExprTuple	65
60	ExprRange	lambda_map: 66 start_index: 124 end_index: 125
61	ExprRange	lambda_map: 67 start_index: 124 end_index: 125
62	Operation	operator: 69 operands: 68
63	Operation	operator: 69 operands: 70
64	ExprRange	lambda_map: 71 start_index: 124 end_index: 125
65	Operation	operator: 72 operands: 73
66	Lambda	parameter: 106 body: 97
67	Lambda	parameter: 106 body: 98
68	ExprTuple	74, 75
69	Literal
70	ExprTuple	76, 77
71	Lambda	parameter: 106 body: 78
72	Literal
73	ExprTuple	99, 125
74	Operation	operator: 80 operands: 79
75	Operation	operator: 80 operands: 81
76	ExprRange	lambda_map: 82 start_index: 124 end_index: 125
77	ExprRange	lambda_map: 83 start_index: 124 end_index: 125
78	IndexedVar	variable: 130 index: 106
79	ExprTuple	84, 85
80	Literal
81	ExprTuple	86, 87
82	Lambda	parameter: 106 body: 88
83	Lambda	parameter: 106 body: 89
84	Variable
85	Operation	operator: 91 operand: 95
86	Variable
87	Operation	operator: 91 operand: 96
88	Operation	operator: 110 operands: 93
89	Operation	operator: 110 operands: 94
90	ExprTuple	95
91	Literal
92	ExprTuple	96
93	ExprTuple	97, 99
94	ExprTuple	98, 99
95	Lambda	parameter: 132 body: 100
96	Lambda	parameter: 132 body: 101
97	IndexedVar	variable: 116 index: 106
98	IndexedVar	variable: 117 index: 106
99	Literal
100	Conditional	value: 103 condition: 105
101	Conditional	value: 104 condition: 105
102	ExprTuple	106
103	Operation	operator: 108 operands: 107
104	Operation	operator: 108 operands: 109
105	Operation	operator: 110 operands: 111
106	Variable
107	ExprTuple	112, 114
108	Literal
109	ExprTuple	113, 114
110	Literal
111	ExprTuple	132, 115
112	IndexedVar	variable: 116 index: 132
113	IndexedVar	variable: 117 index: 132
114	Operation	operator: 118 operands: 119
115	Operation	operator: 120 operands: 121
116	Variable
117	Variable
118	Literal
119	ExprTuple	122, 123
120	Literal
121	ExprTuple	124, 125
122	Operation	operator: 126 operand: 129
123	Operation	operator: 127 operand: 129
124	Literal
125	Variable
126	Literal
127	Literal
128	ExprTuple	129
129	IndexedVar	variable: 130 index: 132
130	Variable
131	ExprTuple	132
132	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 ¶