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, V, W, i, j, m, n
from proveit.core_expr_types import v_1_to_m, v_i, w_1_to_n, w_j
from proveit.linear_algebra import OrthoNormBases, VecSum
from proveit.logic import Equals, Forall, InSet
from proveit.numbers import Interval, one
from proveit.physics.quantum import Bra, Ket, Qmult
from proveit.physics.quantum.algebra import v_1_to_m_kets, w_1_to_n_kets

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [v_1_to_m], instance_expr = Forall(instance_param_or_params = [w_1_to_n], instance_expr = Equals(A, VecSum(index_or_indices = [i], summand = VecSum(index_or_indices = [j], summand = Qmult(Bra(w_j), A, Ket(v_i), Ket(w_j), Bra(v_i)), domain = Interval(one, n)), domain = Interval(one, m))), condition = InSet(w_1_to_n_kets, OrthoNormBases(W))).with_wrapping(), condition = InSet(v_1_to_m_kets, OrthoNormBases(V))).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_{m}~|~\left\{\lvert v_{1} \rangle, \lvert v_{2} \rangle, \ldots, \lvert v_{m} \rangle\right\} \in \textrm{O.N.Bases}\left(V\right)}~\\
\left[\begin{array}{l}\forall_{w_{1}, w_{2}, \ldots, w_{n}~|~\left\{\lvert w_{1} \rangle, \lvert w_{2} \rangle, \ldots, \lvert w_{n} \rangle\right\} \in \textrm{O.N.Bases}\left(W\right)}~\\
\left(A = \left(\sum_{i=1}^{m} \left(\sum_{j=1}^{n} \left(\langle w_{j} \rvert \thinspace A \thinspace \lvert v_{i} \rangle \thinspace \lvert w_{j} \rangle \thinspace \langle v_{i} \rvert\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: 9 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: 80 end_index: 60
6	Operation	operator: 9 operand: 12
7	Operation	operator: 64 operands: 11
8	Lambda	parameter: 61 body: 42
9	Literal
10	ExprTuple	12
11	ExprTuple	13, 14
12	Lambda	parameters: 15 body: 16
13	Operation	operator: 34 operands: 17
14	Operation	operator: 36 operand: 23
15	ExprTuple	19
16	Conditional	value: 20 condition: 21
17	ExprTuple	22
18	ExprTuple	23
19	ExprRange	lambda_map: 24 start_index: 80 end_index: 81
20	Operation	operator: 25 operands: 26
21	Operation	operator: 64 operands: 27
22	ExprRange	lambda_map: 28 start_index: 80 end_index: 60
23	Variable
24	Lambda	parameter: 61 body: 54
25	Literal
26	ExprTuple	67, 29
27	ExprTuple	30, 31
28	Lambda	parameter: 61 body: 32
29	Operation	operator: 48 operand: 39
30	Operation	operator: 34 operands: 35
31	Operation	operator: 36 operand: 41
32	Operation	operator: 72 operand: 42
33	ExprTuple	39
34	Literal
35	ExprTuple	40
36	Literal
37	ExprTuple	41
38	ExprTuple	42
39	Lambda	parameter: 87 body: 43
40	ExprRange	lambda_map: 44 start_index: 80 end_index: 81
41	Variable
42	IndexedVar	variable: 84 index: 61
43	Conditional	value: 45 condition: 46
44	Lambda	parameter: 61 body: 47
45	Operation	operator: 48 operand: 52
46	Operation	operator: 64 operands: 50
47	Operation	operator: 72 operand: 54
48	Literal
49	ExprTuple	52
50	ExprTuple	87, 53
51	ExprTuple	54
52	Lambda	parameter: 86 body: 55
53	Operation	operator: 76 operands: 56
54	IndexedVar	variable: 82 index: 61
55	Conditional	value: 58 condition: 59
56	ExprTuple	80, 60
57	ExprTuple	61
58	Operation	operator: 62 operands: 63
59	Operation	operator: 64 operands: 65
60	Variable
61	Variable
62	Literal
63	ExprTuple	66, 67, 68, 69, 70
64	Literal
65	ExprTuple	86, 71
66	Operation	operator: 74 operand: 78
67	Variable
68	Operation	operator: 72 operand: 79
69	Operation	operator: 72 operand: 78
70	Operation	operator: 74 operand: 79
71	Operation	operator: 76 operands: 77
72	Literal
73	ExprTuple	78
74	Literal
75	ExprTuple	79
76	Literal
77	ExprTuple	80, 81
78	IndexedVar	variable: 82 index: 86
79	IndexedVar	variable: 84 index: 87
80	Literal
81	Variable
82	Variable
83	ExprTuple	86
84	Variable
85	ExprTuple	87
86	Variable
87	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 ¶