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 U, i, j, n
from proveit.core_expr_types import v_1_to_n, v_i, v_j
from proveit.linear_algebra import Adj, MatrixSpace, OrthoNormBases, Unitary
from proveit.logic import CartExp, Equals, Forall, Implies, InSet, Set
from proveit.numbers import Complex, Interval, KroneckerDelta, 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
expr = Forall(instance_param_or_params = [U], instance_expr = Forall(instance_param_or_params = [v_1_to_n], instance_expr = Implies(Forall(instance_param_or_params = [i, j], instance_expr = Equals(Qmult(Bra(v_i), Adj(U), U, Ket(v_j)), KroneckerDelta(i, j)), domain = Interval(one, n)), InSet(U, Unitary(n))), condition = InSet(Set(v_1_to_n_kets), OrthoNormBases(CartExp(Complex, n)))).with_wrapping(), domain = MatrixSpace(field = Complex, rows = n, columns = n))

# 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_{U \in \mathbb{C}^{n \times 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(\mathbb{C}^{n}\right)}~\\
\left(\left[\forall_{i, j \in \{1~\ldotp \ldotp~n\}}~\left(\left(\langle v_{i} \rvert \thinspace U^{\dagger} \thinspace U \thinspace \lvert v_{j} \rangle\right) = \delta_{i, j}\right)\right] \Rightarrow \left(U \in \textrm{U}\left(n\right)\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: 25 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 79 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 25 operand: 8
5	Operation	operator: 61 operands: 7
6	ExprTuple	8
7	ExprTuple	79, 9
8	Lambda	parameters: 10 body: 11
9	Operation	operator: 12 operands: 13
10	ExprTuple	14
11	Conditional	value: 15 condition: 16
12	Literal
13	NamedExprs	field: 45 rows: 82 columns: 82
14	ExprRange	lambda_map: 17 start_index: 81 end_index: 82
15	Operation	operator: 18 operands: 19
16	Operation	operator: 61 operands: 20
17	Lambda	parameter: 83 body: 68
18	Literal
19	ExprTuple	21, 22
20	ExprTuple	23, 24
21	Operation	operator: 25 operand: 31
22	Operation	operator: 61 operands: 27
23	Operation	operator: 38 operand: 33
24	Operation	operator: 29 operand: 34
25	Literal
26	ExprTuple	31
27	ExprTuple	79, 32
28	ExprTuple	33
29	Literal
30	ExprTuple	34
31	Lambda	parameters: 59 body: 35
32	Operation	operator: 36 operand: 82
33	Operation	operator: 38 operands: 39
34	Operation	operator: 40 operands: 41
35	Conditional	value: 42 condition: 43
36	Literal
37	ExprTuple	82
38	Literal
39	ExprTuple	44
40	Literal
41	ExprTuple	45, 82
42	Operation	operator: 46 operands: 47
43	Operation	operator: 48 operands: 49
44	ExprRange	lambda_map: 50 start_index: 81 end_index: 82
45	Literal
46	Literal
47	ExprTuple	51, 52
48	Literal
49	ExprTuple	53, 54
50	Lambda	parameter: 83 body: 55
51	Operation	operator: 56 operands: 57
52	Operation	operator: 58 operands: 59
53	Operation	operator: 61 operands: 60
54	Operation	operator: 61 operands: 62
55	Operation	operator: 73 operand: 68
56	Literal
57	ExprTuple	64, 65, 79, 66
58	Literal
59	ExprTuple	87, 88
60	ExprTuple	87, 67
61	Literal
62	ExprTuple	88, 67
63	ExprTuple	68
64	Operation	operator: 69 operand: 78
65	Operation	operator: 71 operand: 79
66	Operation	operator: 73 operand: 80
67	Operation	operator: 75 operands: 76
68	IndexedVar	variable: 85 index: 83
69	Literal
70	ExprTuple	78
71	Literal
72	ExprTuple	79
73	Literal
74	ExprTuple	80
75	Literal
76	ExprTuple	81, 82
77	ExprTuple	83
78	IndexedVar	variable: 85 index: 87
79	Variable
80	IndexedVar	variable: 85 index: 88
81	Literal
82	Variable
83	Variable
84	ExprTuple	87
85	Variable
86	ExprTuple	88
87	Variable
88	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 ¶