Expression of type Conditional

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 Conditional, 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, 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
expr = Conditional(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)), InSet(n, NaturalPos))

expr:

# 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())

\left\{\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] \textrm{ if } n \in \mathbb{N}^+\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 29 operand: 5
2	Operation	operator: 65 operands: 4
3	ExprTuple	5
4	ExprTuple	86, 6
5	Lambda	parameter: 83 body: 7
6	Literal
7	Conditional	value: 8 condition: 9
8	Operation	operator: 29 operand: 12
9	Operation	operator: 65 operands: 11
10	ExprTuple	12
11	ExprTuple	83, 13
12	Lambda	parameters: 14 body: 15
13	Operation	operator: 16 operands: 17
14	ExprTuple	18
15	Conditional	value: 19 condition: 20
16	Literal
17	NamedExprs	field: 49 rows: 86 columns: 86
18	ExprRange	lambda_map: 21 start_index: 85 end_index: 86
19	Operation	operator: 22 operands: 23
20	Operation	operator: 65 operands: 24
21	Lambda	parameter: 87 body: 72
22	Literal
23	ExprTuple	25, 26
24	ExprTuple	27, 28
25	Operation	operator: 29 operand: 35
26	Operation	operator: 65 operands: 31
27	Operation	operator: 42 operand: 37
28	Operation	operator: 33 operand: 38
29	Literal
30	ExprTuple	35
31	ExprTuple	83, 36
32	ExprTuple	37
33	Literal
34	ExprTuple	38
35	Lambda	parameters: 63 body: 39
36	Operation	operator: 40 operand: 86
37	Operation	operator: 42 operands: 43
38	Operation	operator: 44 operands: 45
39	Conditional	value: 46 condition: 47
40	Literal
41	ExprTuple	86
42	Literal
43	ExprTuple	48
44	Literal
45	ExprTuple	49, 86
46	Operation	operator: 50 operands: 51
47	Operation	operator: 52 operands: 53
48	ExprRange	lambda_map: 54 start_index: 85 end_index: 86
49	Literal
50	Literal
51	ExprTuple	55, 56
52	Literal
53	ExprTuple	57, 58
54	Lambda	parameter: 87 body: 59
55	Operation	operator: 60 operands: 61
56	Operation	operator: 62 operands: 63
57	Operation	operator: 65 operands: 64
58	Operation	operator: 65 operands: 66
59	Operation	operator: 77 operand: 72
60	Literal
61	ExprTuple	68, 69, 83, 70
62	Literal
63	ExprTuple	91, 92
64	ExprTuple	91, 71
65	Literal
66	ExprTuple	92, 71
67	ExprTuple	72
68	Operation	operator: 73 operand: 82
69	Operation	operator: 75 operand: 83
70	Operation	operator: 77 operand: 84
71	Operation	operator: 79 operands: 80
72	IndexedVar	variable: 89 index: 87
73	Literal
74	ExprTuple	82
75	Literal
76	ExprTuple	83
77	Literal
78	ExprTuple	84
79	Literal
80	ExprTuple	85, 86
81	ExprTuple	87
82	IndexedVar	variable: 89 index: 91
83	Variable
84	IndexedVar	variable: 89 index: 92
85	Literal
86	Variable
87	Variable
88	ExprTuple	91
89	Variable
90	ExprTuple	92
91	Variable
92	Variable