Expression of type ExprTuple

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, ExprTuple, Lambda, 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 = ExprTuple(Lambda(U, Conditional(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(), InSet(U, MatrixSpace(field = Complex, rows = n, columns = n)))))

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(U \mapsto \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} \textrm{ if } U \in \mathbb{C}^{n \times n}\right..\right)

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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